Emergent Phenomena in Quantum Many Body Systems

Lincoln D. CarrDepartment of Physics, Colorado School of Mines, Golden, CO 80401, USA

(Dated: March 27, 2009)

Everyone is familiar with single particle quantum mechanics from course work: we all learn aboutquantum tunneling through a barrier, probability waves, and the Hydrogen atom as undergradu-ates, and move on as graduate students to path integrals, the Dyson series, and density matrixformalism. But what are the quantum properties of many particles? Is it enough to write downthe symmetrization postulate and decide that the Universe is divided into bosons and fermions, andderive a quantum version of statistical mechanics? Or, is there more to the story? In this course,we will explore the new, emergent properties that appear in quantum many body systems, fromquantum phase transitions to vortices in quantum hydrodynamics. While the subject matter willcomprise a broad overview, I will also present some specific solution methods for hard quantummany body problems. I hope these methods can be useful to students in solving real, pressingresearch problems, not just textbook examples.

2

Contents

I. Introduction 4

II. Quantum Mechanics 4A. The Postulates 4B. The Qubit 6C. Randomness, Entanglement, and Partial Traces 7

1. The State Matrix 72. Putting Qubits Together 10

D. Infinite Systems: Second Quantization and a Very Brief Introduction to Quantum Field Theory 121. A Single Quantum Harmonic Oscillator 122. The Classical Lattice in One Dimension 133. The Quantum Lattice in One Dimension 144. The Continuum Limit and Quantum Field Theory 15

III. The Quantum Ising Model 17A. What is a Quantum Phase Transition? 17B. Statement of the Hamiltonian and Qualitative Analysis 18C. Perturbative Treatment of Strong and Weak Coupling 19D. Exact Spectrum and the Jordan-Wigner Transformation 21E. Scaling Transformations 22

IV. The Quantum Rotor Model 25A. Statement of the Hamiltonian and Physical Motivation 25B. Weak and Strong Coupling 27C. Continuum Theory and the Semiclassical Spin Limit 28D. Zero Temperature Critical Point 29

1. Quantum Paramagnetic Phase 292. Critical Point 303. Magnetically Ordered Phase 31

V. Hubbard Models 33A. Physical Motivation 33B. Bose-Hubbard Model 33

1. First Mean Field Approach: Gutzwiller Ansatz 332. Second Mean Field Approach: Hubbard-Stratanovich Transformation 36

C. Fermi-Bose Hubbard Hamiltonian 37D. Molecular Hubbard Hamiltonian 41

1. Quantum Measures 422. Case Study: Hard Core Bosonic Molecules at Half Filling 43

VI. Semiclassical Theory and Quantum Hydrodynamics 49A. Uniform Condensate 52B. Quantum Hydrodynamic Equations 53C. Vortex Dynamics in Two Dimensions 54

VII. Numerical Method I: Pseudo-Spectral Adaptive Time-step Runge Kutta 55A. Solitons and stationary states 55

1. Mathematical form 552. Physical interpretation 573. Phase engineering of dynamics 58

B. One dimension 581. Stability 582. Expectations in higher dimensionality 61

C. Quasi-One-Dimension 611. Connection with Experiments 612. Stability 623. Manipulation Via Phase 64

3

D. Beyond quasi-one-dimension 641. Vortex creation 652. Vortex interaction 673. Shock and phase rigidity 68

VIII. Numerical Method II: Exact Diagonalization 68A. Bose-Hubbard Hamiltonian in a Rotating Frame 68B. Physical Measurable Quantities: Observables 70C. Test Cases 71D. General Filling: The Many Body Problem 72

IX. Numerical Method III: Time Evolving Block Decimation 73A. Quantum Many-Body Theory 73B. Discrete Mean-Field Theory From Discrete Quantum Many-Body Theory 74C. Time-Evolving Block Decimation Algorithm 75

1. The Vidal Decomposition 752. Two-Site Operation 753. Real Time Evolution 764. Sources of Error and Convergence Properties 76

D. Constrained Imaginary Time Relaxation in DNLS 771. Fundamental Dark Soliton Solutions 772. Density and Phase Engineering of Gray Solitons 77

E. Time Evolution of Quantum Solitons 771. Standing Solitons 772. Soliton-Soliton Collisions 78

X. Appendix: Simple Exercises to Learn Exact Diagonalization 79A. Bose-Hubbard Model 79B. Our Hilbert Space: Fock Space 79C. Generating Matrix Representations of Operators 80D. Exercises 80

References 82

4

I. INTRODUCTION

Is physics really all about billiard balls? Yes, in quantum mechanics we find that the billiard balls are somehowfuzzy, but we still knock them into each other (scattering theory) and catch them in pockets (detectors). On theother hand, we know from experiments such as the Stern-Gerlach experiment that quantum mechanical particlesshow interference patterns like waves – thus the supposed wave-particle controversy. Still, even in the Stern-Gerlachapparatus, in which one streams spin-1/2 particles through a magnetic field and measures the ensuing deviation intheir trajectories, the concepts are based on measuring a single particle at a time. After many, many measurementsof spin-1/2 particles, one by one, we build up an interference pattern at the detector, and speak of waves.

What if we could measure many particles at once? Would we find the same kinds of wave-like phenomena? Wouldwe simply have a stronger signal at the detector, or might there be new features that arise in a many-body quantumsystem? In this course, I will discuss emergent phenomena in such systems. Some key questions we will address are:

• What is complexity, and what do we mean by emergent phenomena in complex systems?

• What unique perspective on complex systems does quantum mechanics offer?

• How can quantum mechanics, which is linear, appear nonlinear?

The term “emergent phenomena” appears in multiple fields of science. We call a pattern emergent if we cannotpredict it based on properties of single constituents of a system. For example, in neuroscience there is a fairly clearunderstanding of what a neuron is and how it operates. However, many neurons in the brain give rise to consciousness;consciousness cannot be predicted from the properties of a single neuron. Consciousness can be considered as anemergent phenomena of the neurons in the brain. We say that a collection of neurons is a complex system because

it gives rise to new patterns: a complex system is one which exhibits collective behavior which is not predictablefrom its constituents. Moreover, we can say that, in some sense, consciousness is a macroscopic property of theensemble of microscopic (mesoscopic: 10 to 50 µm) neurons. Thus the concept of macroscopic patterns, particularlypersistent patterns, is tied to the term “emergent phenomena.” Other macroscopic everyday examples include thehive intelligence of an ant colony and the great red spot on Jupiter.

In this course, we will study emergent phenomena in the context of interacting quantum many body systems. Todo this, I will describe key macroscopic properties of quantum many body systems. One such set of properties isdescribed by the theory of quantum phase transitions. Universality, scaling, critical exponents, symmetry, and theorder parameter are all important aspects of quantum phase transitions. Specifically, in Sections III, IV and V wewill study the Ising model, the quantum rotor, and Hubbard Hamiltonians, all paradigmatic models. Our discussionrelies on the standard text by Sachdev [1], as well as recent review articles [2] and our own work in this area [3–5].Another set of important properties we can address are solitons and vortices. These particle-like collective excitationsof quantum many body systems can be derived as solutions of quantum hydrodynamics, a mean field, or semiclassicaltheory. In Sec. VI I will present quantum hydrodynamics and work through some specific problems. Here I draw onthe review by Fetter and Svidzinsky [6], standard texts [7], and our own work [8–15]. Finally, quantum many bodyproblems are complex and therefore frequently require numerical methods. In Sections VII, VIII, and IX, I provide anoverview of three such methods, together with physical applications. In the appendix I provide some simple exercisesfor working towards one of these methods. I refer the reader to the Algorithms and Libraries for Physics Simulationsproject for further reading and substantial codes [16, 17].

I want to state from the outset that I acknowledge members of my research group and scientific collaborators whocontributed both to my understanding and to sources for these notes [5, 10–12, 18, 19]. These persons include RajivBhat, Joachim Brand, Hilary Brown, Charles W. Clark, Murray Holland, Nathan Kutz, Mary Ann Leung, RyanMishmash, William P. Reinhardt, and Michael Wall. I acknowledge support from the National Science Foundationunder Grant PHY-0547845 as part of the NSF CAREER program.

I begin with an overview of the theory of quantum mechanics, including density matrix formalism, second quanti-zation, and all the other basic elements needed for the rest of the course.

II. QUANTUM MECHANICS

A. The Postulates

Let us begin with the postulates of quantum mechanics for pure states [20–23]. I note that these postulates dependheavily on the concepts of linear vector spaces, in particular dual vector spaces, which can be represented as row andcolumn vectors in linear algebra, with operators as matrices.

5

1. States: The state vector |φ〉 completely specifies the properties of a quantum system. If |χ〉 and |φ〉 are bothelements of the dual vector space, then

|ψ〉 =λ|φ〉 + µ|χ〉

||λ|φ〉 + µ|χ〉|| (1)

is also an element of the same space and represents a physical space. (This is the principle of superposition.)

2. Probability: If |φ〉 is the state of the system, and |χ〉 is some other state, then the probability of finding |φ〉 in|χ〉 is given by

P (φ→ χ) =|〈χ|φ〉|2

||χ||2||φ||2 , (2)

where ||φ||2 ≡ 〈φ|φ〉 is the normalization.

3. Operators: With every physical property there exists an associated Hermitian operator A which acts in the

space of states |ψ〉. Because |ψ〉 is a linear vector space, and A is diagonalizable,

A =∑

n

an|n〉〈n| (3)

in some orthonormal basis |n〉, where the weights an ∈ R and |n〉 are the eigenvectors of A.

Hidden in this postulate is the concept of measurement. To find the average, or expectation value of Awith respect to state |ψ〉, one measures A many times for identically prepared states |ψ〉. Then

〈ψ|A|ψ〉 = limN→∞

1

N

N∑

p=1

Ap, (4)

where Ap is the result of the pth measurement and N is the number of times the experiment is performed.

4. Dynamics: Time evolution is described by

i~d

dt|φ(t)〉 = H(t)|φ(t)〉, (5)

where H(t) is the Hamiltonian. Equation (5), for a single particle and in position representation, is theSchrodinger equation. The Hamiltonian is taken as Hermitian, meaning norm-preserving. For this reason thispostulate is sometimes called “unitary time evolution.” For a Hamiltonian H which commmutes at differenttimes, [H(t), H(t′)] = 0, the unitary operator for time evolution is:

|ψ(t2)〉 = U(t2, t1)|ψ(t1)〉, (6)

U(t2, t1) ≡ exp

[

− i

~

∫ t2

t1

dt H(t)

]

. (7)

5. Symmetrization: Particles are divided into two categories, called bosons and fermions. The state of bosons isalways symmetric under permutation of two particles and the spin of bosons is an integer multiple of ~. Thestate of fermions is always antisymmetric under permutation of two particles and the spin of fermions is ahalf-integer multiple of ~. There are no partially symmetric states. We write this as

Pkl|i(1)1 i(2)2 · · · i(k)k i

(l)l · · · 〉 = ±|i(1)1 i

(2)2 · · · i(k)l i

(l)k · · · 〉, (8)

where Pkl exchanges the kth and lth particles. In Eq. (8), the subscripts refer to the particle label and thesuperscripts to the particle state.

It is sometimes stated in quantum field theory books that the symmetrization postulate depends on CPTsymmetry (charge, parity, time-reversal). This is incorrect [24, 25].

6

What can we expect from the postulates? Having seen these postulates for the first time, would we have imaginedquantum tunneling? How about a quantum computer? Certainly we can prove important features of quantummechanics, such as the Heisenberg uncertainty relation, in a straighforward manner:

〈(∆A)2〉〈(∆B)2〉 ≥ 1

4|〈[A, B]〉|2, , (9)

where 〈(∆A)2〉 ≡ 〈A2〉 − 〈A〉2.Let us recall some outstanding features of quantum mechanics at the graduate level. Although we can speak of

abstract quantum mechanics, as for instance in the proof of Eq. (9), in order to have a clear discussion I choose severalpractical examples.

B. The Qubit

The simplest quantum mechanical system is the two-state system, called qubit or spin-1/2. Recall the Stern-Gerlachdevice (sketch on board). Suppose our oven produces a zero temperature pure state |ψ〉, one silver atom at a time.Each atom is measured to have spin sz± = ±~/2, where the atom’s angular deviation up or down after passingthrough a magnetic field gradient determines its spin. In terms of actual measurement, I emphasize that a probabilitydistribution is only known after a large number of measurements, and always has statistical uncertainty (error bars)associated with it. We find

pψ(sz+) ≡ |〈sz+|ψ〉|2 =N+

N, (10)

pψ(sz−) ≡ |〈sz−|ψ〉|2 =N−N

, (11)

(12)

where N± is the number of times the atoms are measured to deviate up/down, and N is the total number of

measurements. We can represent Hermitian operators S corresponding to the experimentally observed quantity ofspin (as measured by angular deviation up or down) in terms of Pauli matrices:

σx ≡(

0 11 0

)

,

σy ≡(

0 −ii 0

)

,

σz ≡(

1 00 −1

)

, (13)

(14)

where S ≡ ~

2 σ.We visualize the state of each atom in the experiment in terms of the Bloch sphere, as shown in Fig. 1. For

example, |ψ〉 = |sz+〉 or “spin-up in the z-direction” corresponds to the north pole of the Bloch sphere; |ψ〉 = |sz−〉or “spin-down in the z-direction” corresponds to the south pole. Likewise, |sx±〉 and |sy±〉 lie on the equator alongthe indicated axes. Despite this being an apparently 3D system, any one axis spans the state space:

|ψ〉 = c1|sz+〉 + c2|sz−〉 , (15)

where c1, c2 ∈ C. The Pauli operators defined in Eq. (14) are given in the Sz basis, which is why the σz operatoris diagonal. One could equivalently choose an Sx or Sy basis, using a similarity transformation to diagonalize theappropriate Pauli operator. In the language of qubits we take

|0〉 ≡ |sz+〉 = north pole of Bloch sphere ,

|1〉 ≡ |sz−〉 = south pole of Bloch sphere . (16)

Then the generic state of a qubit can be written in the form

|ψ〉 = cos(θ/2)|0〉 + sin(θ/2)eiφ|1〉 , (17)

7

FIG. 1: The Bloch sphere: the upper (lower) pole |0〉 (|1〉) corresponds to spin up (down) in the z-basis. A classical bit hasonly these two poles, while a quantum bit covers the whole surface of the sphere.

where the angles are as indicated in Fig. 1. There are only two real quantities, despite the four apparent in c1, c2,because of two constraints. First, the overall state is only defined up to an arbitrary phase. Second, we requirenormalization,

||ψ||2 = 〈ψ|ψ〉 = 1 . (18)

Last, we emphasize that what we mean mathematically by “spans the state space” is a complete basis, summarizedin the completeness relation:

∑

α∈0,1|α〉〈α| = 1 , (19)

where 1 means the unity operator, or the unit matrix.The meaning of the name “qubit” is apparent from this discussion and Fig. 1. Quantum mechanics allows for a

superposition of classical bit 0,1, which can be parameterized in terms of the surface of a sphere in 3D. Thus a quantumcomputer consists of an array of spheres, graphically speaking; in Sec. III we will seek the emergent properties ofmany such spheres, in terms of the quantum Ising model.

C. Randomness, Entanglement, and Partial Traces

1. The State Matrix

There are really two sources of randomness or probability in quantum mechanics. The first is the probabilitydistribution of the pure state wavefunction, as already discussed in our statement of the postulates in Sec. II A. Thesecond is loss of information in the form of a mixed state, of which a thermal distribution is a special case. We canmotivate this second form of randomness by consideration of the Stern-Gerlach experiment. The oven is supposed toproduce a random beam which we can separate out into spin-up and spin-down, i.e., |0〉 and |1〉. So let’s suppose theoven made a state |ψ〉 = 1√

2|0〉+ 1√

2|1〉. This seems to be an even superposition of 0 and 1. But if we measure in the

x-direction instead of the z-direction we find

σx|ψ〉 = 1|ψ〉 . (20)

8

Thus ψ is an eigenstate, and we always measure the state at a particular point on the equator of the Bloch sphere. Wewill always find the state points somewhere on the Bloch sphere. How do we get a truly random state, independentof basis?

The answer can be constructed in terms of the density matrix, invented by von Neumann near the beginning ofquantum mechanics in 1927. In this formalism we replace vectors in Hilbert space with a special operator, sometimescalled the state operator or state matrix, defined as

ρ ≡∑

α

pαPα , (21)

where the projectors Pα ≡ |α〉〈α|. The most general form of the density matrix is

ρ ≡∑

i,j

pi,j |i〉〈j| , (22)

but we can always find a basis |α〉 in which it is diagonal. This matrix has nothing to do with density in the senseof a density measurement – therefore I use the term “state matrix” for the rest of our discussion.

The state matrix has the following mathematical properties:

1. ρ = ρ†, that is, the density matrix is Hermitian;

2.∑

α pα = 1, a normalization requirement;

3. the density operator formalism reproduces the state ket (vector) formalism completely.

We interpret pα as a probability that the system is in ket |α〉. Note that the state matrix is Hermitian, so pα ∈ R;and the normalization ensures that the total probability is 1. It follows that Tr(ρ) = 1.

How can we connect this new concept to physically measurable quantities? Consider A = A†. Then the expectationvalue, or average is

〈ψ|A|ψ〉 =∑

j,k

〈ψ|j〉〈j|A|k〉〈k|ψ〉

=∑

j,k

〈k|ψ〉〈ψ|j〉〈j|A|k〉

=∑

j,k

〈k|Pψ|j〉〈j|A|k〉

=∑

k

〈k|PψA|k〉

= Tr(PψA) , (23)

where we have used the completeness relation of Eq. (19). It follows from the definition of the density operator that

〈A〉 =∑

α

Tr(pαPαA)

= Tr(ρA) . (24)

Equation 24 encompasses the state ket or vector formalism in the special case where pψ = 1, pα = 0 for all α 6= ψ.We call this a pure state. Pure states have the special property that

Tr(ρ2) = 1 . (25)

I encourage the reader to prove this.We can now produce a maximally random state from the Stern-Gerlach oven. We claim such a state is

ρ =1

2|0〉〈0| + 1

2|1〉〈1| , (26)

i.e., an outer product of the north pole of the Bloch sphere added to a second outer product of the south pole, andevenly weighted. The reader can prove that 〈σ〉 = 0, so that in whatever direction the states are measured, theaverage result is always zero.

9

Between a pure state, which has no randomness (besides the inherent “randomness” of the probability distribution),and a maximally random mixture, there are many possible probability distributions of kets |α〉. One such distributionis a thermal distribution

pα =e−βEα

Z, (27)

β ≡ 1

kBT, (28)

Z =∑

α

e−βEα . (29)

(30)

For a thermal probability distribution, ρ is diagonal in the energy eigenbasis: H |α〉 = Eα|α〉, where H is the “energyoperator” called the Hamiltonian.

A measure of randomness is given by the von Neumann entropy, a generalization of the Gibbs/Shannon entropy

Sclassical ≡ −∑

j

pj log pj . (31)

The Gibbs/Shannon entropy is a special case of a denumerably infinite set of entropies called Renyi entropies :

Sq ≡1

1 − qlog

n∑

j=1

pqj

, (32)

where in the limit q → 1, Sclassical is recovered. The Renyi entropies manipulate a vector probability pi, and can becharacterized as q-point correlations. The von Neumann entropy is a function of a 2-tensor or matrix probability, i.e.,the state matrix:

SvN ≡ −Tr (ρ log ρ) . (33)

This quantifies our word “randomness.” There is a similiar generalization of the other Renyi entropies to quantumRenyi entropies, where a trace is used to deal with the matrix probability object. (Note that we could derive thethermal probability distribution by maximizing the von Neumann entropy subject to a constraint by the method ofLagrange multipliers, just as it is done in statistical mechanics.)

From Eq. 27,

ρ =∑

α

e−βEα

ZPα . (34)

Since |α〉 are energy eigenstates, H |α〉 = Eα|α〉, and

[H ]α =

E0

E1

. . .ED−1

, (35)

where the square brackets specify the basis |α〉 and we have taken the Hilbert space to be D dimensions (for thequbit, D = 2). Then we can write the state matrix as

ρ =e−βH

Z, (36)

Z ≡ Tr(

e−βH)

. (37)

So, for a thermal probability distribution,

〈A〉 =TrAe−βH

Tr e−βH, (38)

10

Let’s check our concepts of randomness in terms of the limits of the state matrix. In the limit T → 0, if we takethe ground state energy E0 = 0, e−βEα = 0 for all α 6= 0 and limT→0 e

−βE0 = 1 for α = 0. Then

limT→0

[ρ]α =

10

. . .0

, (39)

a pure state. In the opposite limit T → ∞, e−βEα = 1 for all α. Then one can show that Z = D, and

limT→∞

[ρ]α =

1/D1/D

. . .1/D

=1

D1 , (40)

Let’s check the entropy. In the zero temperature limit

limT→0

SvN = −Trρ log ρ = −Tr

10

. . .

0

log

10

. . .

0

= 0 . (41)

In the infinite temperature limit,

limT→0

SvN = −Trρ log ρ = −Tr

[

1

D1 log

(

1

D1

)]

= −Tr

[(

1

Dlog

1

D

)

1

]

= logD . (42)

I note that the base of the log function is taken as 2 in quantum information and e in statistical mechanics.A thermal distribution is only really meaningful for a large dimensional Hilbert space. What do we mean by the

“temperature” of a single qubit? We simply mean an exponential fit to two points, which could just as well be fitby a line, or any other two-parameter function. In fact, non-thermal probability distributions are just as commonas thermal ones in modern experiments. We can obtain general probability distributions by considering a subspaceof a full Hilbert space. A natural physical circumstance which requires the state matrix formalism is a many body

quantum system. I begin our discussion of such systems with just 2 qubits.

2. Putting Qubits Together

Two qubits span four dimensions in Hilbert space. I use the notation H(D) to refer to a Hilbert space of Ddimensions. Note that the term “Hilbert space” in mathematics is normally reserved for D = ∞, but in physics weuse the term to refer to the dual linear vector space underlying quantum mechanics, irrespective of dimension. We

take H(2)A spanned by |0〉A, |1〉A as the Hilbert space describing states of qubit A, and H(2)

B spanned by |0〉B, |1〉Bas the Hilbert space describing states of qubit B. Then the full Hilbert space for the two particles is H(4) = H(2)

A ⊗H(2)B .

We now introduce the notion of a direct product of state kets. Suppose

|ψ〉A ∈ H(2)A ,

|ψ〉B ∈ H(2)B . (43)

Then

|ψ〉A ⊗ |ψ〉B ∈ H(4) = H(2)A ⊗H(2)

A . (44)

The Pauli operators σA and σB then act independently on |ψ〉A and |ψ〉B :

σA → σA ⊗ 1B , (45)

σB → 1A ⊗ σB , (46)

11

and

σA ⊗ 1B|ψ〉A ⊗ |ψ〉B = (σA|ψ〉A) ⊗ |ψ〉B , (47)

1A ⊗ σB |ψ〉A ⊗ |ψ〉B = |ψ〉A ⊗ (σB|ψ〉B) . (48)

We will often write |ψ〉A ⊗ |ψ〉B as |ψ〉A|ψ〉B for shorthand. However, for operators we must explicitly write the ⊗symbol to be clear that we are constructing a larger matrix, rather than simply multiplying matrices.

The question then arises, are all two particle states |ψ〉 ∈ H(4) a direct product of one particle states? That is,can we always write |ψ〉 = |ψ〉A ⊗ |ψ〉B for some |ψ〉A and |ψ〉B? This is analogous to the question, can all functionsf(x, y) be written as f(x, y) = fA(x)fB(y)? We call this concept separability. To answer this, we’ll slightly simplifyour notation. Let |00〉 ≡ |0〉A ⊗ |0〉B, |01〉 ≡ |0〉A ⊗ |1〉B, etc. Then the basis γ ≡ |00〉, |01〉, |10〉, |11〉 spans H(4).Note that this is just binary for |0〉, |1〉, |2〉, |3〉.

Consider an arbitrary pure state |ψ〉 ∈ H(4). Then

|ψ〉 =∑

j

cj |j〉 = c0|00〉 + c1|01〉 + c2|10〉 + c3|11〉 . (49)

On the other hand, for |ψ〉A ∈ H(2)A and |ψ〉B ∈ H(2)

B ,

|ψ〉A = a0|0〉A + a1|1〉A , (50)

|ψ〉B = b0|0〉B + b1|1〉B , (51)

where aj , bj, cj ∈ C. Suppose |ψ〉 = |ψ〉A ⊗ |ψ〉B . Then

a0b0|00〉 + a0b1|01〉 + a1b0|10〉 + a1b1|11〉 = c0|00〉 + c1|01〉 + c2|10〉 + c3|11〉 . (52)

This can only be true for c0c3 = c1c2, which is not the general case. Therefore not all states of H(4) are separable as

a single direct product of states in H(2)A and H(2)

B . QED.We call states for which c0c3 6= c1c2 entangled. Let’s consider an important example, the spin singlet:

|ψ〉 =1√2

(|01〉 − |10〉) . (53)

What are the consequences of such an entangled state? The state matrix for this pure state is ρ = |ψ〉〈ψ|. To askabout the properties of the A and B qubits separately, we need the reduced state matrix (also called the reduceddensity matrix) for each particle:

ρA ≡ TrB ρ , (54)

ρB ≡ TrAρ , . (55)

The symbol TrB is called the partial trace. The direct product allowed us to increase the size of a matrix; the partialtrace allows to decrease the size of a matrix in a self-consistent way. In fact, it can be proven that it is the unique wayto reduce a Hilbert space to a subspace if we require that expectation values behave appropriately in that subspace.The partial trace is defined as follows:

TrB(|ψA〉〈ψA| ⊗ |ψB〉〈ψB |) ≡ |ψA〉〈ψA|Tr(|ψB〉〈ψB |) (56)

and likewise for TrB, where we have moved the subspace subscripts A and B into the kets for notational convenience.In matrix notation and for the state matrix,

[ρA]nAmA≡∑

nB

[ρ]nAnB ,mAnA(57)

where nA,mA run over the basis for the A system/qubit and nB runs over the basis for the B system/qubit.Let’s work through a specific example, namely, the spin singlet. Suppose we measure only the A qubit without

measuring the B qubit. What kind of state would we find? The answer is obtained via the partial trace. First, wedetermine the density matrix:

ρ =

[

1√2

(|01〉 − |10〉)] [

1√2

(〈01| − |〈10|)]

=1

2(|01〉〈01| − |10〉〈01| − |01〉〈10|+ |10〉〈10|) . (58)

12

In matrix representation:

[ρ] =

0 0 0 00 1 −1 00 −1 1 00 0 0 0

(59)

Let’s perform the partial trace:

ρA = TrB ρ =1

2[TrB(|01〉〈01|) − TrB(|10〉〈01|) − TrB(|01〉〈10|) + TrB(|10〉〈10|)] . (60)

Using the key relation

Tr(|a〉〈b|) = 〈b|a〉 , (61)

we find

ρA =1

2[|0〉〈0|〈1|1〉 − |1〉〈0|〈1|0〉 − |0〉〈1|〈0|1〉 + |1〉〈1|〈0|0〉] . (62)

Then by orthonormality,

ρA =1

2(|0〉〈0| + |1〉〈1|) =

1

21 , (63)

or in matrix representation,

[ρA] =1

2

(

1 00 1

)

, (64)

a maximally mixed state!Therefore, although we know everything about the joint state of the two qubits (pure state), we have a complete

loss of information, or maximal randomness of a single qubit (maximally mixed). This is a hallmark of entangledsystems: information about particle A is required to obtain complete information about particle B. They are notindependent.

In many body systems, we will deal with a lattice of qubits, or higher dimensional objects called qudits. Even thoughthe whole system is a pure state at zero temperature, nevertheless we can speak of mixed states for subsystems (andtherefore an effective finite temperature, if we wish to fit the eigenvalues of the reduced state matrix to an exponential).

Consideration of few state systems leads us to elementary quantum chemistry, for instance benzene. We find thatelectrons prefer to be delocalized on the benzene ring. The states the electrons occupy are described by the theoryof point symmetry groups, making a connection to abstract algebra. In the limit in which the number of sites in thering approaches infinity, we obtain the lattice problem and an associated band theory. We’ll next treat the case of aninfinite number of sites.

D. Infinite Systems: Second Quantization and a Very Brief Introduction to Quantum Field Theory

1. A Single Quantum Harmonic Oscillator

I begin with reminding the reader of the harmonic oscillator system. In 1D the Hamiltonian is

H = ~ω

(

a†a+1

21

)

, (65)

where the annihilation (creation) operator a (a†) destroys (creates) a quantum of energy. The number operator is

N ≡ a†a. These operators act on state |n〉 as

a|n〉 =√n|n− 1〉 , (66)

a†|n〉 =√n+ 1|n+ 1〉 , (67)

N |n〉 = n|n〉 , (68)

13

and satisfy the fundamental commutation relations

[a, a†] = 1 , [a, a] = [a†, a†] = 0 , (69)

The classical Hamiltonian for the harmonic oscillator is

Hclassical =p2

2m+

1

2kx2 , (70)

with k = mω the spring constant and m the mass of the particle subject to restoring force F = −kx. The quantumHamiltonian follows from Dirac’s prescription,

x→ x , p→ p , [x, p]P.B. →1

i~[x, p] , (71)

where P.B. stands for Poisson brackets, plus the identification of annihilation and creation operators in terms of theposition and momentum operators:

a ≡√mω

2~

(

x+i

mωp

)

,

a† ≡√mω

2~

(

x− i

mωp

)

. (72)

2. The Classical Lattice in One Dimension

Let us now consider a classical 1D lattice consisting of a chain of N masses m connected by springs of springconstant K. The springs represent the coupling between “sites”, i.e., masses. Suppose that the equilibrium positionsof the masses are given by xn = nℓ (that is, in equilibrium they are each spaced ℓ apart), with phase space coordinatesqn, pn = mqn. Let them satisfy periodic boundary conditions xn+N = xn. This is a familiar classical model withHamiltonian

Hcl =

N−1∑

n=0

p2n

2m+

1

2K

N−1∑

n=0

(qn+1 − qn)2 . (73)

The equation of motion is

mqn = −K [(qn − qn−1) + (qn − qn+1)] . (74)

This is a wave equation discretized in space but not in time, since the right hand side is a discretized second derivative.We seek the normal modes of Hcl. To this end it is convenient to perform Fourier transforms on qn, pn:

qk =1√N

N−1∑

n=0

eikxnqn ≡∑

n

Uknqn , (75)

where Ukn is a unitary matrix and qk is the Fourier transform of qn. One can prove that U †U = UU † = 1. Let’srewrite the interval 0 ≤ k ≤ 2π(N − 1)/(Nℓ) so that it is centered around k = 0: −π

ℓ ≤ k ≤ πℓ . Note that I’ve

assumed N ≫ 1. Then this looks like the first Brillouin zone from band theory. The inverse Fourier transform is

qn =1√N

π/ℓ∑

k=−π/ℓe−ikxnqk =

∑

k

U †nkqk =

∑

k

U−knqk , (76)

where I note that

U †nk = U∗

kn = U−kn , (77)

and∑

n

UknU†nk′ =

∑

n

UknU−k′n = δkk′ . (78)

14

Substituting qn → pn, qk → pk one obtains the same Fourier transforms for momentum.We proceed to use these discrete Fourier transforms to evaluate the Hamiltonian. The kinetic energy is

∑

n

p2n =

∑

n

∑

k,k′

U−knU−k′n pkpk′ =∑

k,k′

δk,−k′ pkpk′ =∑

k

pkp−k . (79)

The potential energy is

∑

n

(qn+1 − qn)2 =

∑

n

∑

k,k′

[

(

e−ikℓ − 1)

(

e−ik′ℓ − 1

)

U−knU−k′n qkqk′]

=∑

k

(

e−ikℓ − 1) (

eikℓ − 1)

qkq−k = 4∑

k

sin2

(

kℓ

2

)

qkq−k . (80)

Putting these together,

Hcl =1

2m

∑

k

pkp−k +K

2

∑

k

4 sin2

(

kℓ

2

)

qkq−k , (81)

or

Hcl =1

2m

∑

k

pkp−k +m

2

∑

k

ω2kqkq−k , (82)

where

ωk ≡ 2

√

K

msin

( |k|ℓ2

)

(83)

is the frequency of the kth mode and all modes are decoupled except ±k. Equation (83) is the dispersion relationfor this classical system (sketch on board). Note that the dispersion relation is linear for small k, like photons:E = pc⇒ ~ω = ~kc⇒ ω ∝ k. For small k,

ωk =

√

K

mℓ|k| + O(k2) , (84)

and one can identify c =√

K/mℓ as an effective velocity.

3. The Quantum Lattice in One Dimension

We can solve the analogous quantum problem using Dirac’s prescription from Eq. (71). Then

[qn, pn′ ] = i~δn,n′ 1 . (85)

Only operators corresponding to the same site have a non-zero commutation relation. Then the quantum Hamiltonianis

H =1

2m

∑

k

pkp−k +m

2

∑

k

ω2k qk q−k , (86)

in k-space. Using our Fourier transforms we can obtain the commutation relation in k-space:

[qk, pk′ ] =∑

n,n′

UknUk′n′ [qn, pn′ ] = i~1∑

n

UknUk′n = i~δk,−k′ 1 . (87)

Finally, we can decouple the modes k and −k via new operators which we define as

qk =

√

~

2mωk

(

ak + a†−k

)

, (88)

pk = −i√

~mωk2

(

ak − a†−k

)

, (89)

15

Substituting Eqs. (88)-(89) into Eq. (87), we obtain

H =

π/ℓ∑

k=−π/ℓ~ωk

(

a†kak +1

21

)

. (90)

Equation (90) is now just a sum of independent harmonic oscillators!

What do ak and a†k do to a state? Let |r〉 be an energy eigenstate, H|r〉 = Er|r〉. Then

Hak|r〉 = (akH + [H, ak])|r〉 = (Er − ~ωk)ak|r〉 , (91)

as can be calculated from [ak, a†k′ ] = δk,k′ 1. Similarly, Ha†k|r〉 = (Er + ~ωk)a

†k|r〉. Therefore ak, a

†k destroy/create

an excitation with a quantum of energy ~ωk. We call these elementary excitations quasiparticles or phonons. The

operator Nk ≡ a†kak measures the number of phonons in mode k. Note that [H, Nk] = 0. We can define |0k〉 as the

ground state of the kth mode: ak|0k〉 = 0. Then normalized states with nk phonons in the kth mode are

|nk〉 =1√nk!

(

a†k

)nk

|0k〉 , (92)

and the eigenstates of H are

|r〉 = |nk1〉 ⊗ |nk2〉 ⊗ · · · ⊗ |nkN 〉 , (93)

where each element of k1, . . . , kN takes on quantized values from −π/ℓ to π/ℓ. The eigenvalues of |r〉 can then becalculated:

H |r〉 =

π/ℓ∑

k=−π/ℓ

(

nk +1

2

)

~ωk

|r〉 . (94)

The Hilbert space defined by

|r〉 = |nk1〉 ⊗ |nk2〉 ⊗ · · · ⊗ |nkN 〉 | nk1 , . . . , nkN ∈ |Z| (95)

is called Fock space. Sometimes we also call it number space, informally. We’ll use Fock space in various formsthroughout our treatment of quantum phase transitions.

4. The Continuum Limit and Quantum Field Theory

If we take the limit in which the mass-spring system becomes continuous, we obtain an elastic spring in onedimension, or an elastic membrane in two. The easiest way to obtain this limit is to consider only long wavelengthexcitations. Then the discrete nature of the system is never resolved, i.e., |k|ℓ ≪ 1, with the excitation wavelengthλ ≡ 2π/k, and the dispersion relation is

ωk = 2

√

K

msin

|k|ℓ2

→√

K

m|k|ℓ ≡ csk . (96)

Here cs is the effective sound speed, as before. It is convenient to rewrite this [23] in term’s of a Young’s modulus

Y [98], cs ≡√

Y/µ, where µ ≡ m/ℓ is the linear mass density and Y = Kℓ. We define the continuum fields

φ(xn, t) = qn(t) , π(xn, t) =1

ℓpn(t) . (97)

We can use finite differencing to approximate their derivatives, since ℓ→ 0:

∂φ

∂x

∣

∣

∣

∣

x=xn

=φ(xn+1, t) − φ(xn, t)

ℓ, (98)

∂2φ

∂x2

∣

∣

∣

∣

x=xn

=φ(xn+1, t) − 2φ(xn, t) + φ(xn−1, t)

ℓ2. (99)

16

The equation of motion then becomes the wave equation:

µ∂2φ

∂t2=Y

ℓ2[φ(xn+1) − φ(xn)] + [φ(xn−1) − φ(xn)] → ∂2φ

∂t2= c2s

∂2φ

∂x2. (100)

Similarly, the discrete classical Hamiltonian

Hcl = ℓ∑

n

π2(xn)

2µ+

1

2Kℓ

[

φ(xn+1) − φ(xn)

ℓ

]2

, (101)

becomes an integral:

Hcl =

∫ L

0

dx

[

π2(x)

2µ+

1

2µc2s

(

∂φ

∂x

)2]

, (102)

where I’ve taken the limits ℓ→ 0, N → ∞, L = Nℓ =const.; m→ 0, µ = m/ℓ =const.; and K → ∞, Y = Kℓ =const.I again mix φ(x) and π(x) into normal modes via a Fourier transform, this time a continuous one:

φk = −φ∗−k ≡ 1√L

∫ L

0

dx eikxφ(x) = limℓ→0

ℓ√Nℓ

∑

n

eikxnφ(xn) =√ℓqk . (103)

The inverse Fourier transform is

φ(x) =1√L

∑

k

e−ikxφk . (104)

Analogous relations hold for π(x), leading to πk. We note that the finite length of the string makes the inverse Fouriertransform a sum; for an infinite string we would expect an integral over momentum as well.

Let us proceed to quantize this scalar field defined on the continuous spatial interval [0, L] via the Dirac prescription:

φk → φk , πk → πk , (105)

and

[φk, πk′ ] = i~δk,−k′ 1 , (106)

in analogy to the discrete commutation relation. Following the same hybridization procedure as for the discrete case,

H =∑

k

(

1

2µπkπ−k +

1

2c2sk

2φkφ−k

)

. (107)

As before, the operator substitution

φk =

√

~

2µωk

(

ak + a†−k

)

, (108)

πk =

√

~µωk2

(

ak − a†−k

)

, (109)

diagonalizes the Hamiltonian:

H =∑

k

~ωk

(

a†kak +1

21

)

, (110)

where ak and a†k are annihilation and creation operators for quasiparticles in momentum state k. Recall ωk = csk.The complete basis for the Hilbert space is now

|r〉 = |nk1〉 ⊗ |nk2〉 ⊗ · · · | nk1 , nk2 , . . . ∈ |Z| (111)

17

and the index j in nkj ranges from 1 to ∞, rather than being truncated at N as before. This leads to an apparentultraviolet divergence in the ground state energy

∞∑

k=kmin

1

2~csk → ∞ , (112)

but this is an artifact of the way we set up our continuum, and can be moderated by an appropriate cut-off, e.g., anon-zero lattice const. such that kmax ∼ 2π/ℓ. Recall that we assumed long wavelength excitations. In our study ofquantum phase transitions we will often utilize such a cut-off, which we will call Λ = kmax.

The field operators φ(x) and π(x) can be obtained via the Fourier transform, and one can show that

[φ(x), π(x′)] = i~δ(x− x′)1 . (113)

Note that x and x′ are parameters, not operators. This formalism is called second quantization.

III. THE QUANTUM ISING MODEL

A. What is a Quantum Phase Transition?

Quantum phase transitions are driven by quantum fluctuations, just as classical phase transitions are driven bythermal fluctuations. We look for a parameter g analogous to temperature which can drive such transitions; we willchoose this parameter to be dimensionless. In particular, we focus on second order quantum phase transitions, forwhich we expect a critical exponent as

∆ ∼ J |g − gc|zν , (114)

where ∆ is the characteristic energy scales of fluctuations near critical point gc, and zν is a critical exponent. Thefact that these fluctuations vanish as g → gc is what makes this a second order transition. The value of zν is usuallyuniversal, i.e., independent of the microscopic details of the Hamiltonian; systems with the same critical exponent arecalled a universality class. In addition to the vanishing energy scale, there is a length scale ξ which diverges as

ξ−1 ∼ Λ|g − gc|ν , (115)

where Λ ∼ 1/a with a the lattice constant, or distance between sites.Equations (115) and (114) are true strictly for the ground state at zero temperature. From the starting point of

T = 0 and g ∼ gc one can understand low but finite temperature properties in real experimental systems, a subjectwe encourage the reader to pursue; in our discussion, we restrict ourselves to T = 0 for the sake of brevity.

To illustrate these ideas, we will consider two Hamiltonians which have useful classical limits: the quantum Isingand quantum rotor models. A quantum Ising model is a quantized version of the well known classical Ising model

H = −J∑

〈i,j〉SiSj , (116)

where Si and Sj are ±1 and the notation 〈i, j〉 means the sum is over nearest neighbors on a lattice. Taking whiteto be +1 and black to be -1, one can show that at high temperature a lattice of spins looks like “white noise”; atlow temperature the system freezes into a black or white phase. Since all-black and all-white are degenerate, this isan example of spontaneous symmetry breaking. The classical phase transition actually only occurs in two and higherdimensions. The Ising model is exactly solvable by easy methods in 1D, by harder methods in 2D, and numericallyin D = 3 and higher dimensions.

A one-dimensional classical model maps onto a zero-dimensional quantum model, otherwise known as a single site.This is one motivator for building a quantum computer: a single quantum element, be it a qubit or a qudit (a qudit isa higher but finite dimensional quantum object like a quantum rotor), maps onto an infinite one-dimensional classicalsystem. If a single quantum element contains an entire infinite classical system, what can one do with an array ofsuch elements?

18

B. Statement of the Hamiltonian and Qualitative Analysis

In one spatial dimension the quantum Ising model is

HI = −Jg∑

i

σxi − J∑

〈i,j〉σzi σ

zj . (117)

In Eq. (117) J > 0 is an exchange constant which sets the single-spin energy scale, the σx,zi are the Pauli operators asdefined in Eq. (14), and g > 0 is a dimensionless parameter which tunes the system across a quantum phase transition,as we will explain. As in Sec. II D, the “hats” refer to operators, and operators on different sites i commute. Wenote that there is one additional term in Eq. (117) as compared to Eq. (116); this extra term represents quantumtunneling events which flip the spin in the z direction. In a physical systems such as a ferromagnetic material, J mightrepresent the magnetic coupling between nearest neighbors (actually a truncated dipole-dipole interaction), while gcould represent a transverse magnetic field. In quantum information, we can visualize each site as a qubit.

There are two straightforward limits we can consider: g ≫ 1 and g ≪ 1. In the strong coupling, or g ≫ 1 case, theground state is

|g.s.〉 =⊗

i

|0x〉i , (118)

to leading order in 1/g, where |0x〉i = 1√2(|0z〉i ± |1z〉i), as discussed in Sec. II B. All spins are on the equator of the

Bloch sphere – see Fig. (1). Equation (118) is a single Fock state, called a product state. It has no spatial entanglementbetween sites. If we use perturbation theory to make corrections in 1/g, we will find that, for large g, correlations areshort-ranged:

〈g.s.′|σzi σzj |g.s.′〉 ∼ e−|xi−xj|/ξ , (119)

where xi is the spatial location of site i, ξ is the correlation length, and |g.s.′〉 is now with respect to all orders ofperturbation in 1/g.

In the weak coupling, or g ≪ 1 limit, we again find a product state:

|g.s.(0)〉 =⊗

i

|0z〉i . (120)

However, this time all qubits are locked into the north pole of the Bloch sphere (spin up in the z direction). Again,this is a single Fock state, and there is no spatial entanglement. In fact, this state is degenerate with

|g.s.(1)〉 =⊗

i

|1z〉i : (121)

thus the system can be all-north or all-south, just as in the classical case discussed in Sec. III A. What happens as g isincreased away from zero? Increasing g mixes in a few sites with the opposite pole; however, the degeneracy remains,due to an exact global Z2 symmetry transformation generated by the unitary operator

⊗

i σxi . That is to say, for

finite but small g, mostly-north and mostly-south are degenerate. A study in perturbation theory of correlations inthe north-south observable σz yields

lim|xi−xj|→∞

〈g.s.|σzi σzj |g.s.〉 = N20 , (122)

where N0 is called the magnetization. We find that N0 = 1 for g = 0 while N0 is less than unity but still non-zero asg is increased to below gc.

To understand why the quantum phase transition happens, we observe that Eq. (118) can not be mapped con-tinuously and analytically onto Eq. (120) or Eq. (121) as a function of g, because the correlators Eq. (119) andEq. (122) cannot be mapped onto each other. We call the point where this jump between two different behaviors inthe correlator occurs, the critical point. We call the state for g > gc a quantum paramagnet and the state for g < gca magnetically ordered state (here a ferromagnet).

Although we will not discuss finite temperature or excited states in general in our overview, we mention brieflythat, in the infinite system limit, the excitation spectrum has a gap for g 6= gc, and is gapless for g = gc; a gap refersto the fact that a finite, non-zero amount of energy is needed to reach the first excitation above the ground state. Fora finite system this is always true (think of the particle in a box problem); for an infinite system it need not be true.

19

C. Perturbative Treatment of Strong and Weak Coupling

In one spatial dimension the quantum Ising model in a transverse field is

HI1 = −J∑

i

(

gσxi + σzi σzi+1

)

. (123)

We will study this case in some detail. Before, we argued on a mainly qualitative basis that there must exist aquantum critical point at gc. Let us now determine the exact value of gc and properties of the quantum Ising chainfor g near gc, starting from a more careful consideration of strongly and weakly interacting limits.

For g = ∞ the first excited state can be written exactly:

|i〉 = |1x〉i⊗

j 6=i|0x〉j , (124)

where the state is countably infinitely degenerate. In this state, one site is flipped from one side of the Bloch-sphereequator to the other, from east to west. We call these single particle states. The next degenerate set of excited statesinvolve two such flips, then three, up to n-particle states. What is the effect of perturbations in 1/g? To lowest order,non-infinite g lifts the degeneracy. For example, for single-particle states, the exchange term σzi σ

zi+1 in Eq. (123) is

not diagonal in the basis of “equator” states; this leads to the coupling

〈i|HI1|i+ 1〉 = −J . (125)

This “hops” the single particle excitation between sites. Analogous to our discussion of Sec. II D, we diagonalize thesecond-quantized Hamiltonian by transforming to momentum space:

|k〉 =1√N

∑

j

eikxj |j〉 , (126)

where N is the number of sites. We find the eigenstate has energy

ǫk = Jg

[

2 − 2

gcos(ka) + O(

1

g2)

]

, (127)

where we have set the zero of energy to the ground state. Recall that a is the lattice constant. Then the lowest energyof a single-particle state is ǫ0 = 2gJ − 2J .

We can now use a similar idea for two-particle states. The exact two particle states for g = ∞ are

|i, j〉 = |1x〉i ⊗ |1x〉j⊗

m 6=i,j|0x〉m ; (128)

note that |i, j〉 = |j, i〉, so these two-particle states are bosons, and we need only consider i > j. Again, to leadingorder in 1/g the Hamiltonian Eq. (123) couples |i, j〉 to |i±1, j〉 and |i, j±1〉, for all i > j+1. Assuming the particlesare separated by two or more sites, there are no other couplings, and the total energy is just the sum of the twosingle-particle energies: Ek = ǫk1 + ǫk2 , with total momentum k = k1 + k2. When the two single-particle excitationsapproach each other there is an additional mixing in which |i, i− 1〉 also couples to |i+ 1, i− 1〉 and |i, i− 2〉. Thereare no two-particle bound states, as can be verified, and so this becomes a scattering theory problem. Momentum isconserved up to the reciprocal lattice vector Λ ∼ 2π/a. Therefore the two-particle state has the general form

[

ei(k1xi+k2xj) + Sk1k2ei(k2xi+k1xj)

]

|i, j〉 , (129)

where we have considered only small momenta k1, k2 ≪ 2π/a so that momentum is conserved. The factor Sk1k2 isthe S-matrix. You can work directly through the Schrodinger equation problem for two-particle scattering to order1/g and find that Sk1k2 = −1. It can also be shown that this result holds for higher orders in 1/g.

Since k1 and k2 can take on a range of values, so long as k = k1 + k2, there is in fact a continuum of two-particleenergies. The lowest energy is at 2ǫ0. In general, for n-particle states, the lowest energy is nǫ0. If we were to go tohigher orders in 1/g, we would find that n- and (n + 2)-particle states are coupled. However, the qualitative natureof the spectrum remains the same: we still find renormalized one-particle states with a definite energy-momentumrelationship (dispersion relation) and renormalized continuums of states for n ≥ 2 with thresholds at nǫ0. One also

20

finds that the particle number is always conserved in scattering processes; this is a result of the nearest-neighborcoupling in the Hamiltonian Eq. (123).

Let us now consider this excitation spectrum in terms of a measurable quantity, the dynamic structure factor. Thedynamic structure factor is given generically as

S(k, ω) ≡∫

dx

∫

dtC(x, t)e−i(kx−ωt) , (130)

where the space-time correlatorC(x, t) is the continuum limit of the lattice space-time correlator, or dynamic two-point

correlation

C(xi, t) ≡ 〈σzi (t)σz0(0)〉 . (131)

In Eq. (131) the Heisenberg picture is assumed, so that the time dependence is in the operators, not the states. Thestructure factor is measured in neutron scattering experiments, for example. Another useful quantity is the dynamicsusceptibility, which is the same as Eq. (130) except that the Fourier transform is in imaginary time τ ≡ it:

χ(k, ωn) ≡∫ 1/T

0

dτ

∫

dxC(x, τ)e−i(kx−ωnτ) , (132)

where ωn = 2πnT , n ∈ Z is an imaginary frequency which can be analytically continued to real frequencies viaiωn → ω + iδ. The fluctuation-dissipation theorem yields

S(kω) =2

1 − eω/TImχ(k, ω) , (133)

a relation we will find useful. We emphasize that the susceptibility is a linear response function, a very reasonablequantity to calculate if one wishes to understand the result of experimental probes.

Inserting the completeness relation over all eigenstates |s〉 of the Hamiltonian Eq. (123) into Eq. (130), we find

S(k, ω) = 2π∑

s

|〈0|σz(k)|s〉|2δ(ω − Es) , (134)

where s = 0 means the ground state and by σz(k) we mean the Fourier transform of the site-dependent Pauli matrixin the continuum limit. The fluctuation-dissipation theorem yields

Imχ(k, ω) =sgn(ω)

2S(k, |ω|) . (135)

We can deduce the structure factor from our consideration of n-particle states above. The lowest frequency contributorto S(k, ω) is the single-particle state with one spin flipped. This has momentum k and frequency ωk, and yieldsS(k, ω) ∼ δ(ω − ǫk). We call this the quasiparticle peak, and its coefficient is called the quasiparticle amplitude. Forg = ∞ this is the only signal seen in a measurement of the structure factor; but for non-infinite g there is mixing asdiscussed above, and a new signal is seen above the three-particle threshold ω = 3ǫ0, a smooth function correspondingto a continuum of states. We note that Eq. (134) is zero for even numbers of particles, so the next band occurs forfive-particle states, etc.

Let us now consider weak coupling, or g ≪ 1, in a more precise manner, as we have already done for strong coupling.The g = 0 ground states are two-fold degenerate (all-north or all-south) and are described by Eq. (121). Adjustingour notation for the correlator as defined in Eq. (131),

lim|x|→∞

C(x, 0) = N20 6= 0 , (136)

with N0 the spontaneous magnetization. Excited states are described in terms of domain walls. For example, if a spinchain is all-north from −∞ to zero and all south beyond that, we would say there is a domain wall at the origin. Atg = 0 the energy is 2J times the number of domain walls. Small nonzero g leads to corrections in a similar manneras the strong coupling limit; domain walls are effective particles which hop and are momentum eigenstates. Theirenergy is

ǫk = J [2 − 2g cos(ka) + O(g2)] . (137)

Perturbation theory in g mixes states that differ by even numbers of particles, and the structure factor is non-zeroonly for states with even number of particles, in contrast to strong coupling, where odd numbers are required. Onefinds that the structure factor has a delta function at k = 0, ω = 0, due to long range order; this comes from theterm in Eq. (134) where s is the ground state. There is no single-particle contribution, and the threshold for thefirst continuum of excitations is the two-particle threshold. This is a special feature of the 1D quantum Ising model,because excitations are domain walls for weak coupling. The S-matrix can be computed for scattering of domainwalls, and once again one find Sk1k2 = −1 to all orders in g.

21

D. Exact Spectrum and the Jordan-Wigner Transformation

We can give an even more precise description of excitations for all coupling by taking advantage of a very convenientmapping from qubits, or spin-1/2 fermions, to spinless fermions. In this mapping, the north pole of the qubit is anoccupied site and the south pole of the qubit is an empty site. Accordingly, we define spinless fermionic annihilation

and creation operators ci and c†i (note there is only a site index, no spin index) in terms of Pauli operators as follows:

σzi = 1 − 2c†i ci . (138)

The “spin-flip” operators σ± are the same as the spinless fermion annihilation and creation operators:

ci = σ+ ≡ 1

2(σxi + iσyi ) , (139)

c†i = σ− ≡ 1

2(σxi − iσyi ) . (140)

This works for a single site. However, we still have the problem that fermionic operators on different sites anticommute,while spin operators on different sites commute. To fix this, Jordan and Wigner extended the definition:

σ+i =

∏

j<i

(

1 − 2c†j cj)

ci , (141)

σ−i =

∏

j<i

(

1 − 2c†j cj)

c†i , (142)

or inversely,

ci =

∏

j<i

σzj

σ+i , (143)

c†i =

∏

j<i

σzj

σ−i . (144)

The reader can check that Eqs. (141)-(144) satisfy the right anticommutation and commutation [ ] relations:

ci, c†j

= δij , (145)

ci, cj =

c†i , c†j

= 0 , (146)[

σ+i , σ

−j

]

= δij σzi , (147)

[

σzi , σ±j

]

= ±2δij σ±i . (148)

To apply this mapping to the Ising model, we make an additional rotation of π/2 about the y-axis of the Blochsphere, so that σz → σx and σx → σ−z . Then the Jordan-Wigner mapping which we insert into Eq. (123) is

σxi = 1 − 2c†i ci , (149)

σzi = −∏

j<i

(

1 − 2c†j cj)(

ci + c†i

)

, (150)

resulting in a transformed quantum Ising Hamiltonian in diagonal form:

H ′I1 = −J

∑

(

c†i ci+1 + c†i+1ci + c†i c†i+1 + ci+1ci − 2gc†i ci + g1

)

, (151)

where 1 is the unity operator. The total number of fermions N ≡∑i c†i ci does not commute with this Hamiltonian.

Since this fermionic number is proportional to the total magnetization in the x-direction, according to Eq. (149), thisjust means that spins can be flipped, as we already discussed in Sec. (III C).

22

Hamiltonians of quadratic form can be diagonalized analytically. This is the quantum many body equivalent ofcompleting the square, called a canonical transformation, or Boguliubov transformation, after the person who firstused it [26]. In detail, we first make a transformation to momentum space, as was introduced in Sec. II D:

ck =1√M

∑

j

cje−ikrj , (152)

where M is the number of sites. Then the Hamiltonian Eq. (151) becomes

H ′′I1 = J

∑

k

2[g − cos(ka)]c†k ck + i sin(ka)(c†−k c†k + c−k ck) − g1

. (153)

The Boguliubov transformation is

γk ≡ uk ck − ivk c†−k , (154)

where uk, vk ∈ R with normalization u2k + v2

k = 1 and u−k = uk and v−k = −vk. The Boguliubov quasiparticle

operators γk, γ†k satisfy the fermionic commutation relations of Eqs. (145) and (146), and can be inverted as

ck = ukγk + ivkγ†−k . (155)

Choosing the right values of uk and vk yields a diagonal Hamiltonian

HIB =∑

k

ǫk

(

γ†kγk −1

21

)

, (156)

where

ǫk = 2J√

1 + g2 − 2g cos k (157)

is the single-particle energy. The values of uk and vk which lead to this diagonalization are

tan θk =sin(ka)

g − cos(ka), (158)

uk = cos(θk/2) , (159)

vk = sin(θk/2) . (160)

It is instructive to interpret this mapping which diagonalizes the quantum Ising Hamiltonian. First, from Eq. (158)it is apparent that g = 1 is a special point in the theory. We can guess that this might be the critical point. Second,for small ka, tan θk ∼ ka/(1− g), so that the main weight in the Boguliubov quasiparticle comes from ck. In general,we observe that the Boguliubov quasiparticle is a superposition of a particle and a hole. We can think of thesequasiparticles as free fermions, or as hard core bosons. We observe that the excitation energy of Eq. (157) is nonzeroand positive for all k except for g = 1. The energy gap is at k = 0 and takes the value 2J |1 − g|. At g = 1 this gapvanishes, another indication that this is the critical point gc we have been seeking. The principle of universality tellsus that there is a universal theory which describes critical properties near gc = 1.

In the following section, we sketch this theory.

E. Scaling Transformations

We transform to a continuum theory near the critical point as follows. First, define the continuum Fermi field

Ψ(xi) =1√aci (161)

satisfying the continuum Fermi operator commutation relations

[Ψ(x), Ψ†(x′)] = δ(x− x′) . (162)

23

Then we can expand the non-diagonal scalar fermion configuration space representation of the quantum Ising Hamil-tonian, Eq. (151), in terms of spatial gradients (via finite differencing):

HC = E0 −∫

dx

[

c

2

(

Ψ† ∂Ψ†

∂x− Ψ

∂Ψ

∂x

)

+ ∆Ψ†Ψ

]

+ · · · , (163)

up to first order gradients, where ∆ = 2J(1− g) and c = 2Ja. To obtain this limit we took J → ∞, g → 1, and a→ 0

while holding Ψ, ∆, and c fixed. At the critical point g = gc = 1 we have ∆ = 0; in the strong coupling quantumparamagnetic phase we have ∆ < 0; and in the weak coupling magnetically ordered phase we have ∆ > 0.

For the scaling analysis we’ll need a quantum field theoretic path integral formulation. I won’t review path integralsin quantum field theory, but I hope you are at least familiar with path integrals in single particle quantum mechanics.If not, bear with me, and try to get the basic idea of what scaling is about, without worrying about the mathematics.In any case, I am going to use a slightly more advanced version of path integrals than you’ve likely seen before ingraduate coursework. We’ll briefly use temperature but then go directly to the T = 0 case.

The partition function Z = Tre−HC/T for a Lagrangian LC written in terms of continuum Fermi operators iswritten in terms of a Grassman path integral (a special formulation invented in the 50’s to deal with path integrationfor fermions) as

Z =

∫

DΨDΨ† exp

(

−∫ 1/T

0

dτdxLC)

, (164)

where Ψ and Ψ† are now complex Grassman fields over space x and imaginary time τ and

LC = Ψ† ∂Ψ

∂τ− c

2

(

Ψ† ∂Ψ†

∂x− Ψ

∂Ψ

∂x

)

− ∆Ψ†Ψ . (165)

Equation (165) is the universal theory describing critical points in the quantum Ising model. By universal we meanthat if we modified the details of our model, say by introducing next-nearest neighbor coupling, we’d still get backEq. (165); only the values of the gap ∆ and the velocity c would change. It is only the particular model Hamiltonianwe have chosen that has allowed us to derive this exactly.

The new continuum theory can be diagonalized with similar methods to the ones we have already presented. Theresult is an excitation energy which is relativistic in form:

ǫk = (∆2 + c2k2)1/2 . (166)

Then ∆ is the energy gap at zero temperature, and c is the velocity of long wavelength excitations (long wavelengthbecause we are in the continuum limit, ka≪ 1). In fact, this relativistic dispersion tells us that we can formally mapthis theory onto Lorentz-invariant field theory, in particular of massive Majorana fermions.

Why does Eq. (165) represent a universal field theory? We know that universality is tied to scaling transformations.Let’s let Λ ≡ π/a represent the microscopic momentum for the microscopic length a that we want to scale away from.We’d like to be able to eliminate high momentum degrees of freedom from Λ to Λe−ℓ, with ℓ dimensionless. If we cando this, then we have a universal theory which does not depend on its microscopic details but only on long wavelength/ low momentum properties. Since the path integral is Gaussian, if we try to integrate out these high momentummodes we only add an overall constant to the free energy. Then our new LC is valid for length scales greater thanaeℓ. To make this all self-consistent, we have to rescale space, imaginary time, and the Grassman field:

x′ ≡ xe−ℓ , (167)

τ ′ ≡ τe−zℓ , (168)

Ψ′ ≡ Ψeℓ/2 , (169)

where the factor of 1/2 in the last definition is so the normalization will scale correctly. In our theory the dynamic

critical exponent z = 1, but in general such scalings are not necessarily symmetric in space and imaginary time, so zcan be different from 1. Plugging these new definitions in, we find that Eq. (165) has exactly the same form as beforefor ∆ = T = 0, i.e. at the critical point. Note that only for z = 1 is the velocity c invariant under our rescaling.

This invariance under rescaling is a symmetry of the problem. Recall that, for example, behavior of a system underrotations classifies particles as fermions or bosons. The classification of a system under scaling transformations isreally what we mean by a “universality class.” Specifically, how are operators and couplings classified according tothe behavior of the Lagrangian under the scaling transformation of Eqs. (167) to (169)?

24

What happens if we move away from the critical point at ∆ = T = 0? If we change only ∆, but not T , the actionremains invariant only if we now rescale the gap:

∆′ = ∆eℓ . (170)

We can write the effect of repeated scaling transformations as

d∆

dℓ= ∆ . (171)

This is called a flow equation in renormalization group theory. What this equation is describing is what an experi-mentalist sees happen to the gap as the resolution of the measuring device is decreased, rather like zooming out witha camera. Only at the critical point does the gap remain fixed at zero; otherwise, it grows to ±∞. Renormalizationgroup methods seek to study how a system changes as a function of scale, a very natural question to ask when lookingfor emergent properties.

Our discussion leads to the concept of a scaling dimension. The scaling dimension is the power to which the lengthrescaling factor eℓ must be raised to to obtain the scaling transformation. For example, for the gap ∆, which functionsas a coupling constant in Eq. (165), we raise eℓ to the first power in Eq. (170). So the scaling dimension is unity. Wedenote this scaling dimension with square brackets as

dim[∆] = 1 . (172)

Then the critical exponent ν is defined as the inverse of this dimension:

ν =1

dim[∆]. (173)

In particular, we choose ν for the most important parameter which perturbs around a critical point. Here that is ∆.From Eqs. (167)-(169),

dim[x] = −1 , (174)

dim[τ ] = −z , (175)

dim[Ψ] = 1/2 . (176)

The temperature, since it is an inverse imaginary time, has dim[T ] = z. For the free energy density, recall thatF = −(T/V ) lnZ. By dimensional analysis dim[V ] = Ddim[x], where D is the spatial dimensions of the quantumproblem. It follows that

dim[F ] = d+ z . (177)

The last interesting scaling dimension is that of the order parameter. Although we have not used the term so far,we clarify that the order parameter is a measure of the north-ness or south-ness of the Ising system, σz , since this iswhat clearly differentiates the quantum paramagnetic phase from the quantum magnetically ordered phase. Althoughwe do not calculate this scaling dimension here, we encourage the reader to investigate this calculation on his or herown [1]. The result is

dim[σz ] =1

8. (178)

This cannot be predicted by dimensional analysis; for this reason it is sometimes called an anomalous dimension.More precisely, the anomalous dimension in the difference between the engineering, or units-based dimension, and thescaling dimension. All of these methods were first laid out for classical systems.

What would happen if we were to generalize HC or LC to include other short range terms? There are two relevantkinds of perturbation. The first is to go to higher order gradients in LC , for example

λ1Ψ† ∂

2Ψ

∂x2. (179)

The second is from additional terms we might add to the original Hamiltonian of Eq. (123); such terms should respectthe Z2 symmetry in order to maintain the same universality class. For example, we might consider σxi σ

xi+1. Going

25

through the same transformations as we performed on other parts of the quantum Ising Hamiltonian we find in thecontinuum limit a term of form

λ2Ψ† ∂Ψ†

∂x

∂Ψ

∂xΨ . (180)

If we then calculate the scaling dimension we find dim[λ1] = −1 and dim[λ2] = −2. Because these dimensions arenegative, a flow equation such as Eq. (171) shows that these are irrelevant parameters: at larger and larger scalesthey become smaller and smaller, and therefore less and less important.

The fact that we cannot find any relevant perturbations at ∆ = 0 is what makes LC the universal continuumquantum field theory of the quantum Ising critical point.

Why do we make qualitative arguments based on scaling dimension and renormalization flow equations? This isbecause (a) most quantum many body Hamiltonians are not solvable by any other means, and (b) these are the mostrelevant considerations when we seek emergent properties.

IV. THE QUANTUM ROTOR MODEL

A. Statement of the Hamiltonian and Physical Motivation

The quantum rotor is an extension of the quantized version of the rigid rotor problem. Examples of rigid rotors arethe spinning top, which requires three angles, and a barbell or classical diatomic molecule, which require two angles.The classical Hamiltonian in the latter case is

H =1

2µR2

(

p2θ +

p2φ

sin2 θ

)

, (181)

where µ is the reduced mass, R is the length of the barbell or molecule, and pφ and pθ are canonical momenta.Equation (181) also describes a pendulum which is free to swing in three dimensions. This is a standard problem inundergraduate classical mechanics. We will quantize this Hamiltonian to obtain the single quantum rotor. An arrayof such quantum rotors makes a lattice problem; we will allow more degrees of freedom than two or three angles,corresponding to more components in the orientation and angular momentum vectors. This is hard to visualize forclassical systems because most people cannot easily think in higher than three dimensions. Is such an idea simplya mathematical construct? No, because a quantum rotor in higher dimensions models the physical case of the lowenergy states of a number of closely coupled electrons. We emphasize that even though our quantum rotors will behigher dimensional objects, the dimensionality is in their rotational degrees of freedom; the spatial dimensionality ofthe lattice remains physical: D = 1, 2, or 3.

The general Hamiltonian for a lattice of quantum rotors is

HR =Jg

2

∑

i

L2i − J

∑

〈i,j〉ni · nj , (182)

where Li and ni are N -component unit vectors representing the angular momentum and orientation of the rotor,respectively. We also require that n2

i = 1 so that the rotors cannot “stretch”, i.e., are the quantum equivalent ofrigid rotors. In Eq. (182), the first term is a kinetic energy and has the single-rotor physics; the second term couplesneighboring rotors, in a similar manner to the Ising Hamiltonian, Eq. (117). The classical rigid rotor corresponds tothe N = 3 case; we consider here higher N , even infinite N , as an extension of the original rotor idea to arbitrarydimensions. We say that the rotor lies always on the surface of an N -dimensional hypersphere (“hyper” just meansthat we allow N > 3).

Since n2i = 1, it follows that the momentum pi of the rotor must always be tangent to the surface of the hypersphere.

The commutation relations are

[nα, pβ ] = iδαβ , (183)

where we use Greek subscripts to emphasize that we consider here the internal degrees of freedom, not the sitesi, j: α, β ∈ 1, . . . , N. We also choose units such that ~ = kB = 1. The Hamiltonian Eq. (182) used the angularmomentum instead of the linear momentum:

Lαβ = nαpβ − nβ pα , (184)

26

according to the usual cross product. These operators are the generators of rotations in N dimensions. The angularmomentum operators satisfy the spin commutation relations

[Lα, Lβ] = iǫαβγLγ , (185)

and the other commutators can be derived accordingly:

[Lα, nβ] = iǫαβγnγ , (186)

[nα, nβ ] = 0 . (187)

As for the Ising model, operators for different site indices commute. For N = 3 and a single site (a single rotor) weknow the solution of this Hamiltonian: it is just the partial waves of the Hydrogen atom. The eigenvalues are

Eℓ =Jg

2ℓ(ℓ+ 1) (188)

with degeneracy 2ℓ+ 1.For the strong coupling, or g ≫ 1 limit, the first term in Eq. (182), the single-rotor kinetic energy, dominates. Then

we expect a quantum paramagnet; the rotors prefer to randomize, yielding a correlation function of form

〈g.s.′|ninj |g.s.′〉 ∼ e−|xi−xj |/ξ . (189)

For weak coupling, or g ≪ 1, we expect a magnetically ordered state with correlation function

lim|xi−xj |→∞

〈g.s.|ninj |g.s.〉 = N20 , (190)

Between these two extremes we expect a second order quantum phase transition, for the same reasons as the Isingmodel. However, the Z2 symmetry will be replaced with O(N) symmetry, as we will explain. It also turns out thatthe story can be different in one spatial dimension and for N = 2. We will focus on spatial dimensions greater than 1and the large N limit for the purposes of the present investigation. We encourage the reader to separately investigateN = 2 and one spatial dimension, which are important and interesting special cases.

In the following, we will consider a slight extension of the quantum rotor Hamiltonian of Eq. (182):

HR =Jg

2

∑

i

L2i − J

∑

〈i,j〉ninj − H ·

∑

i

Li , (191)

where H is a field which couples to the total angular momentum of the lattice Lnet =∑

i Li. As in our discussion ofthe Ising model in Sec. III, when the exchange term proportional to J is dominant the system is in a magneticallyordered state, while the analog of the Ising kinetic energy, proportional to Jg, prefers a quantum paramagnet. In thelatter case we say that quantum fluctuations in the orientation of the order parameter lead to a loss of long-rangeorder. Again, we will find strong-coupling/weak-coupling regimes for large/small g, and anticipate a quantum criticalpoint gc somewhere in the middle. However, unlike for the Ising model, we will not find an overall exact solution.

In order to make a connection with experimental measurements, we consider the dynamic susceptibility and structurefactor. We define the dynamic susceptibility in imaginary time for the order parameter ni, the orientation of theN -component (internal dimension) rotors in d spatial dimensions:

Cαβ(x, τ) ≡ 〈nα(x, τ)nβ(0, 0)〉 , (192)

χαβ(k, ωn) ≡∫ 1/T

0

ddxCαβ(x, τ)e−ik·x−ωnτ , (193)

where n(xi, iτ) is the imaginary time representation of the operator ni. The dynamic structure factor Sαβ(k, ω) isdefined in the same way as Eq. (130), and a fluctuation-dissipation relation connects it to the dynamic susceptibilityin analogy to Eq. (133). For small perturbing field H in Eq. (191) we speak of the uniform susceptibility χu, definedin terms of an expansion in the free energy density F ,

F(H) = F(H = 0) − 1

2χuαβHαHβ . (194)

27

B. Weak and Strong Coupling

For g = ∞, or strong coupling, there is no exchange between sites and Eq. (191) can be diagonalized exactly. Forexample, for N = 3 we find the same structure of eigenstates on each site as if we had an array of non-interactinghydrogen atoms, and wished to know the angular momentum eigenstates:

|g.s.〉 =⊗

i

|ℓ,m〉i , (195)

where ℓ ∈ 0, 1, 2, . . ., −ℓ ≤ m ≤ ℓ, and ℓ,m are independent on each site. The ground state is a quantum paramagnet:

|g.s.〉 =⊗

i

|0, 0〉 . (196)

The analysis is analogous to the Ising model. The lowest excited states are single-particle states in which a single sitehas ℓ = 1. However, we now have a three-fold degeneracy in m. This degeneracy is split by the external field H, andis one reason we chose to treat a slightly extended version of the quantum rotor. We find

ǫkm = Jg

[

1 − 2

3g

∑

µ

cos(kµa) + O(1/g2)

]

−Hm , (197)

where we have chosen the coordinate system so that H = Hez and the sum over µ is over all d spatial dimensions.Comparing this expression to Eq. (127), we observe the extra factor of 3 due to the degeneracy in m.

The strongly interacting limit brings us to a connection between the quantum rotor and a real material, double

layer antiferromagnets. Such a system is realized in spin-ladder compounds in one spatial dimension and double layercompounds in high-temperature superconductors in two dimensions. The Hamiltonian for a system consisting of twolayers of Heisenberg spins S1i and S2i is

Hd = K∑

i

S1i · S2i + J∑

<i,j>

(

S1i · S1j + S2i · S2j

)

, (198)

where the Sni, n the layer index, are spin operators representing the total spin of a set of electrons in localized atomicstates. The spin operators obey the usual commutation relations

[Sαni, Sβn′i′ ] = iǫαβγSγniδnn′δii′ , (199)

where i is the site index. Even though these are the same commutation relations as for the angular momentumLi of the rotors, the portion of the Hilbert space they operate on is restricted to total spin S on each site, i.e.,|S,m〉 |m ∈ −S,−S + 1, . . . , S − 1, S. It follows that S2

ni = S(S + 1)1 for all sites i and layers n.

In the limit K ≫ J in Eq. (198) we can neglect J couplings. Then Hd splits into a pair of decoupled sites. Each sitehas an antiferromagnetic coupling K between spins. In this limit we can diagonalize the double layer Hamiltonian.The spins S1i and S2i add to states on each site of total angular momentum 0 ≤ 2S. Thus the eigenenergies are(K/2)[ℓ(ℓ+ 1) − 2S(S + 1)] with degeneracy 2ℓ+ 1. These are identical to those of the N = 3 quantum rotor exceptfor the upper limit of 2S – so for low energy excitations and strong coupling the two models, Eq. (198) and (191), arethe same. Thus quantum rotors in this case represent pairs of spins. The operators map directly onto one anotheraccording to Li = S1i + S2i; the rotor angular momentum is the total angular momentum of the underlying coupledlayer spin system. The reader can verify that ni = S1i − S2i in the large S limit; the quantum rotor coordinate ni isthe antiferromagnetic order parameter for the spin system. Magnetically ordered states of the rotor model map ontospin states with long range antiferromagnetic order.

Let us turn to weak coupling, g ≪ 1. In this limit consideration of Eq. (191) for H = 0 shows that each rotor prefersto line up in an arbitrary but common direction. Thus there is spontaneous symmetry breaking, as for the quantumIsing model. However, here the degeneracy is continuous, rather than discrete, since any angle ni is allowed. Thenlow energy excitations consist in long wavelength variations of the direction of the rotors on the lattice, where theorientation of neighboring rotors consists of an arbitrarily small angular difference. Because the wavelength can bearbitrarily long, this is a gapless excitation – it can cost an arbitrarily small amount of energy. In this long wavelengthlimit the theory becomes continuous and we can introduce operators π1, π2 canonical to the angular momenta L1, L2

as follows. Let N = 3 and let the ground state be polarized along n = (0, 0, 1). Then excitations are parameterized as

ni = (π1(x, t), π2(x, t), (1 − π21 − π2

2)1/2) , (200)

28

where we have incorporated the “no-stretching” condition on the rotors and |π1|, |π2| have eigenvalues smaller thanunity. Then one can derive the commutation relations

[L1, π2] = i , [L2, π1] = −i . (201)

The Heisenberg equation of motion follows from the postulates of quantum mechanics discussed in Sec. II, i~∂A/∂t =

[H, A]. From the Heisenberg theorem, we derive the linearized equations of motion

∂π1

∂t= JgL2 , (202)

∂L2

∂t= Ja2∇2π1 , (203)

and likewise for π2 and L1. The solution of these equations is two degenerate spin-wave normal modes with dispersionǫk = ck and velocity c = Ja

√g. The two modes are degenerate because there is no preference for the phase to rotate

forwards or backwards. The reason there are two modes (instead of three, as for strong coupling) is because of thebroken symmetry: rotations about 〈n〉 (which we chose in the e3 direction) have no effect. We encourage the readerto quantize the normal modes of Eqs. (202)-(203) via the methods for harmonic oscillators.

What happens if we allow g to be nonzero but still small? Then quantum zero-point motion about the magneticallyordered state enters the picture. As for the Ising model, the magnetization must be reduced by these fluctuations.To lowest order in g:

〈n3〉 = 〈(1 − π21 − π2

2)1/2)〉

≃ 1 − 1

2〈π2

1 + π22〉

= 1 − 1

2

√

gad−1

∫

ddk

(2π)d1

k, (204)

where in the last step we took advantage of the quantized harmonic oscillator solutions of Eqs. (202)-(203). We canimmediately observe that in d = 1 there is a small-k divergence. Although we will not treat the d = 1 quantum rotormodel in this course, we mention briefly that this divergence indicates that there is no magnetic long range order inone spatial dimension; instead, the system is paramagnetic for all g. For d > 1 there is a quantum critical point atsome intermediate value of g.

Of course one can ask what happens if all the terms are retained from the Heisenberg equation of motion, insteadof linearizing. It turns out that this nonlinearity leads to a negligible effect of spin-wave scattering (except again ind = 1, where the effect cannot be neglected). We point out that although the states in the Hilbert space of manybody quantum mechanics adds linearly, operators need not be linear.

C. Continuum Theory and the Semiclassical Spin Limit

For the analysis of this section we again turn to path integration, in particular path integration over imaginarytime. Without the field H, and for d dimensions and N internal spin degrees of freedom, the partition function for arotor of fixed length in the continuum limit is

Z =

∫

Dn(x, τ)δ(n2(x, τ) − 1) exp(−Sn) , (205)

Sn =N

2cg

∫

ddx

∫ 1/T

0

dτ [(∂τn)2 + c2(∇xn)2] . (206)

This is called the O(N) quantum nonlinear sigma model.. The effect of the field H is to cause the rotors to precess.By transforming to the rotating frame, i.e., the frame of the procession, this can be folded into the action as follows:

Z = Tr[exp(−HR/T )] ≃∫

Dn(x, τ)δ(n2 − 1) exp(−Sn) , (207)

Sn =N

2cg

∫

ddx

∫ 1/T

0

dτ [(∂τn − iH× n)2 + c2(∇xn)2] , (208)

29

where the coupling to H is written specifically for N = 3. Because the i in the precession term contributes a complexphase to the weights in the partition function, Eq. (208) has no analog in classical statistical mechanics. The effectivecoupling constant in this path integral description is

g ≡ N√

gad−1 . (209)

The action of Eq. (208) is valid at long distances and times. Therefore we must keep in mind that there is a cutoff athigh momentum Λ ∼ 1/a and frequency ωcut ∼ cΛ. What is the universal physics at length scales much larger thana?

Let us derive the N = ∞ solution in the quantum paramagnetic phase. We can require the non-stretching n2 = 1constraint via a Lagrange multiplier λ. Then

Z =

∫

Dn(x, τ)Dλ(x, τ) exp(−Sn1) , (210)

Sn1 =N

2cg

∫

ddx

∫ 1/T

0

dτ [(∂τn)2 + c2(∇xn)2 + iλ(n2 − 1)] . (211)

Renormalizing n to n ≡√Nn, and noting that Eq. (211) is quadratic in the n, it can be integrated out. Then the

partition function becomes

Z =

∫

Dλ(x, τ) exp

[

−N2

(

Tr ln(−c2∇2 − ∂2τ + iλ) − i

cg

∫ 1/T

0

dτ

∫

ddxλ(x, τ)

)]

, (212)

Since this action has a prefactor of N and we are taking N → ∞, the functional integral is given by its saddle-point value – only the weight at the extremum can remain non-zero. We will take the saddle-point value of λ to beindependent of space and time. Let m2 ≡ iλ. Then the saddle point in terms of parameter m is given by

∫ Λ ddk

(2π)dT∑

ωn

1

c2k2 + ω2n +m2

=1

cg, (213)

where the sum is over Matsubara frequencies ωn ≡ 2nπT , n ∈ Z, as for the Ising model.If we perturb Eq. (211) by putting in a source term, we get the order parameter susceptibility at N = ∞. We state

the result without proof:

χ(k, ω) =cg/N

c2k2 − (ω + iδ)2 +m2, (214)

where χαβ = χδαβ. By expanding the free energy density in powers of H we get obtain the uniform susceptibility

χu = 2T∑

ωn

∫

ddk

(2π)dc2k2 +m2 − ω2

n

(c2k2 +m2 + ω2n)

2. (215)

We have not provided much detail in the derivation of Eqs. (213)-(215). However, the path integration techniquesare standard and can be fleshed out by the reader if need be. Eqs. (213)-(215) are the main results for N = ∞, andwe need them to perform analysis of the three regimes g > gc, g = gc, and g < gc for zero temperature in the nextsection.

D. Zero Temperature Critical Point

1. Quantum Paramagnetic Phase

We can use the relativistic invariance of the action Eq. (208) to our advantage. The summation over Matsubarafrequencies becomes an integral at zero temperature. We introduce space-time momentum p ≡ (k, ω/c). Then thesaddle-point equation Eq. (213) becomes

∫ Λ dd+1p

(2π)d+1

1

p2 + (m/c)2=

1

g, (216)

30

where m is a parameterization from Sec. IVC, not a mass. Let us analyze Eq. (216). The integral increases mono-tonically with decreasing m. As m→ 0 it diverges as ln(1/m) in d = 1, and has a maximum possible finite value ford > 1. Therefore there is always a solution in d = 1, while in higher spatial dimensions there is no solution for someg smaller than some gc, where gc is defined by the m→ 0 limit:

∫ Λ dd+1p

(2π)d+1

1

p2≡ 1

gc, (217)

This is in fact the critical point – below gc, the quantum paramagnet ceases to exist. Although this is apparentlyonly true for N = ∞, in fact it turns out to be true for N > 3 – note that N = 1 is just the Ising model.

To approach the critical point from above, we just subtract Eq. (216) from Eq. (217) to get

1

gc− 1

g≡∫ Λ dd+1p

(2π)d+1

(

1

p2− 1

p2 + (m/c)2

)

. (218)

Even if we take the cutoff Λ → ∞, we still get a finite result for d < 3. Therefore at least if we always make ourmeasurements of physical properties relative to their value at the critical point, the physics is universal. We calld = 3 the upper critical dimension for this reason; it represents a borderline case between universal and non-universal(cutoff-dependent) behavior. We restrict ourselves to d = 2 for the rest of the analysis.

Take Λ → ∞ in Eq. (218). Then the integral yields

1

gc− 1

g= Xd+1

(m

c

)d+1

, (219)

Xd+1 ≡ 2Γ

(

3 − d

2

)

(4π)−(d+1)/2

d− 1, (220)

where Xd is a constant, with Γ the gamma function. We solve this equation for the parameter m, which is related tothe Lagrange multiplier λ. What is the meaning of m? It is the gap: ∆+ ≡ m(T = 0). Recall that there is a gap forthe quantum paramagnet phase while there is no gap for the magnetically ordered phase. We can understand this isa gap from the spectral density Imχ(k, ω); from Eq. (214),

Imχ(k, ω) =cg

N

π

2√

c2k2 + ∆2+

[

δ

(

ω −√

c2k2 + ∆2+

)

− δ

(

ω +√

c2k2 + ∆2+

)]

. (221)

Since this function has weight only at frequencies greater than ∆+, we can understand that it is a gap. The deltafunctions indicate magnon quasiparticles. These magnons are the three-fold degenerate particle we discussed inSec. (IV B) for strong coupling. We can also study the uniform susceptibility from Sec. IVC. We convert thefrequency summation to an integral, since T = 0, and get χu = 0. This is because small H field cannot excite thesystem over the gap.

Finally, we explicitly verify this is a quantum paramagnet phase by observing that equal-time rotor orientationcorrelations (recall that the order parameter is n) decay exponentially:

1

N〈n(x, 0)n(0, 0)〉 =

g

N

∫

dd+1p

(2π)d+1

eip·x

p2 + (∆+/c)2,

=cg

N

1

2c(2π)d/2(∆+/c)(2−d)/2e−x∆+/c

xd/2. (222)

Thus ∆+/c is the inverse correlation length.

2. Critical Point

Armed with these results, we can determine what actually occurs at the critical point gc. From Eq. (219), theenergy gap vanishes as

∆+ ∝ (g − gc)1/(d−1) . (223)

From our discussion of scaling dimension in Sec. III E, it follows that

zν =1

d− 1(224)

31

At the critical point, the equal time rotor orientation correlations decay as

〈n(x, 0)n(0, 0)〉 ∝∫

dd+1p

(2π)d+1

eip·x

p2∝ 1

xd−1. (225)

Since the decay as a function of time has the same exponent, z = 1, as is always the case for a Lorenz-invariant theory.We apply the scaling transformation to Eq. (225) and get dim[n] = (d− 1)/2. This can also be seen by requiring that∫

ddx∫

dτ(∇n)2 in the action be invariant under the scaling transformation.We can use the scaling analysis to determine what the dynamic susceptibility is at the critical point. It is convenient

to parameterize

dim[n] =d+ z − 2 + η

2, (226)

where η is the anomalous dimension that we briefly mentioned in Sec. III E. The anomalous dimension is defined asthe difference between the engineering dimension (what you get just by units analysis) and the actual dimension of theorder parameter. The exact solution of the Ising chain is η = 1/4, as can be proven by more extensive means than wehave time to present here. Recall that the Ising case is an N = 1 rotor. Here, for the N = ∞ case, η = 0. For valuesof N inbetween we can guess that η might be a monotonic function; in fact, if fluctuation corrections are taken intoaccount for finite N this can be proven. The pre-factor of Eq. (221) is called the quasiparticle residue. Let A ≡ gN/c.Equation (226) shows that dim[χ(kω)] = −2 + η; therefore dim[A] = η. It follows that A ∼ (g − gc)

ην . Thus, thereare no quasiparticles, even for η 6= 0, at the critical point. What kind of excitations does one find? Requiring Lorentzinvariance of these excitations, and using scaling dimension analysis, one can show that

χ(k, ω) ∝ 1

(c2k2 − ω2)1−η/2, (227)

at the critical point. Similarly, we can determine χu by scaling analysis of H. In the action Eq. (208), H is coupled toa time-derivative through the square; this is because the only effect of H is to cause the rotors to uniformly precess.This is an exact property, and so must remain under the scaling transformation. Recalling that dim[τ ] = −z anddim[F ] = d+ z, it follows from the definition of χu in Eq. (194) that dim[χu] = d− z.

Thus we have brought the concept of scaling analysis from the initial undergraduate one of “engineering dimension”to that of critical points, where an anomalous dimension must also be taken into account.

3. Magnetically Ordered Phase

Finally, we seek a description of the weak-coupling magnetically ordered phase, where the saddle point requirementis invalid. Previously, we treated every component of n the same. In the magnetically ordered state this is notreasonable, because a particular direction has been selected by symmetry breaking. Therefore we take

n = (√Nr0, π1, . . . , πN−1) , (228)

where we orient the internal coordinate system such that the direction of polarization of the order parameter liesalong e1. We again renormalize the order parameter such that n ≡

√Nn as before. Inserting Eq. (228) into the

action Eq. (208) with Lagrange multiplier λ, and integrating over all components except the polarized direction, i.e.,integrating out π1, . . . , πN−1, we obtain

Z =

∫

DλDr0 exp

[

−N − 1

2Tr ln(−c2∇2 − ∂2

τ + iλ) +iN

cg

∫ 1/T

0

dτ

∫

ddxλ(1 − r20)

]

. (229)

Now we follow our previous procedure to get a saddle point, but in both λ and r0. We again let m2 ≡ iλ. Thespontaneous magnetization is

N0 ≡ 〈n1〉 =√Nr0 . (230)

The saddle point equations are

N20 + g

∫ Λ dd+1p

(2π)d+1

1

p2 + (m/c)2= 1 (231)

m2N0 = 1 , (232)

32

at T = 0. The second saddle point equation requires either N0 = 0, which is the paramagnetic case we know fails forg < gc, or m = 0. The latter requirement is the same as saying the state is gapless. Then we get

N20 = 1 − g

∫ Λ dd+1p

(2π)d+1

1

p2(233)

= 1 − g

gc(234)

according to our earlier definition of the critical point. We can define a critical exponent β for the vanishing of themagnetization according to N0 ∝ (gc − g)β . Then β = 1/2 for N = ∞. Note that in our analysis we took N = N − 1which is only true for N = ∞. Since the scaling dimension of N0 is the same as the scaling dimension of n, it followsfrom Eq. (226) that

2β = (d+ z − 2 + η)ν . (235)

How can we define the approach of the magnetically ordered phase to the critical point? We present a new quantitycalled the spin stiffness ρs. This is a measure of how easy it is to change the orientation of the order parameter.Suppose we wanted the magnetization to precess from the e1 direction to the e2 direction. Then we could define anorder parameter

〈n〉 = N0(cosφ(x), sin φ(x), 0, . . . , 0) , (236)

where φ(x) is a slowly varying function. Since constant phase φ(x) does not change the energy, it is only the gradientof the phase that matters. One can show by symmetry arguments that the lowest order term in the change in theenergy depends on the square of this quantity:

δE =ρs2

∫

ddx(∇φ)2 , (237)

where the change in energy is always with respect to the ground state of a single polarization direction. We nowagain use the scaling transformation to determine the dimension of the spin stiffness. Since φ is an angle, dim[φ] = 0;dim[δE] = z. Therefore

dim[ρs] = d+ z − 2 . (238)

We can then determine the energy scale ∆− for the ground state for weak coupling g < gc based on scaling arguments.This energy scale should have scaling dimension z and physical units of inverse time (since we have taken ~ = 1). Weneed to construct ∆− from powers of ρs, whose scaling dimension is given in Eq. (238) and whose physical dimensionis length2−d × time−1, and from the velocity c, whose scaling dimension is zero and whose physical dimension islength × time−1. The correct (and unique) combination is

∆− ≡(ρsN

)1/(d−1)

c(d−2)/(d−1) , (239)

up to normalization constant N .How can we relate the spin stiffness to a measurable quantity? Let’s define the two-point correlator of spin

components orthogonal to the axis of magnetization. We can take this orthogonal axis in the e2 direction. Then thedynamic susceptibility based on the n2, n2 correlator is

χ⊥(k, ω) =cg/N

c2k2 − (ω − iδ)2. (240)

Suppose we have a slowly varying weak field h(x) which couples linearly to the 2-component of the rotor vector n. Thesystem should follow the slowly varying field. The spin stiffness presents the energy cost in doing so. Mathematically,

δE =

∫

ddx[ρs

2(∇φ)2 − hN0 sinφ

]

. (241)

Minimizing this quantity with respect to variations in φ, we find in Fourier space

〈n2(k)〉 ≃ N0φ(k) =N2

0

ρsk2h(k) , (242)

33

leading to the exact result

limk→0

χ⊥(k, 0) =N2

0

ρsk2. (243)

Combining this with Eq. (234) and the definition of χ⊥(k, ω) we get

ρs = cN

(

1

g− 1

gc

)

. (244)

Therefore at zero coupling the spin is perfectly stiff, while the spin stiffness approaches zero as the system approachesthe critical point.

V. HUBBARD MODELS

A. Physical Motivation

In Sec. V, Sec. VIII, and Sec. IX many of our examples will be taken from cold atoms and molecules in opticallattices. This is because these systems present Hubbard models which can be derived from first principles. However,Hubbard models have been used for a long time to study all kinds of systems, even where they provide only anapproximate picture which cannot be derived from first principles. Originally they were used to treat the motion ofelectrons in transition metals. The Hubbard or Fermi-Hubbard model turns out to be analytically intractable in twoand higher dimensions for repulsively interacting electrons, even in the simplest case of just two energy scales in theHamiltonian: nearest-neighbor hopping, or tunneling, which allows the electrons to move from site to site; and on-siteinteractions, which says that electrons only interact when in the same spatial location.

We’ll start with a model which is solvable, at least by mean field methods [27], the Bose Hubbard model. The BoseHubbard model is deceptively simple; unlike the quantum Ising and quantum rotor models, it has no classical analogin one higher dimension via the quantum-classical field theoretic mapping.

B. Bose-Hubbard Model

1. First Mean Field Approach: Gutzwiller Ansatz

Let bi and b†i be annihilation and creation operators for a boson on site i. Then they obey the bosonic commutationrelations

[bi, b†j] = δij 1 , (245)

with the number operator ni ≡ b†i bi. We choose a Fock-state basis for the Hilbert space. For example, in one spatialdimension the number operator acting on a Fock state yields

ni| . . . , n−1, n0, n1, . . . , ni, . . . 〉 = ni| . . . , n−1, n0, n1, . . . , ni, . . . 〉 , (246)

where ni ∈ |Z| for all i. In the grand canonical ensemble we write the Bose-Hubbard Hamiltonian as

Hb = −t∑

〈i,j〉

(

b†i bj + h.c.)

+U

2

∑

i

ni(

ni − 1)

− µ∑

i

ni . (247)

The coefficients are all energies, which can in fact be derived explicitly from the overlap of Wannier functions ona lattice: t is the hopping or tunneling, where we clarify that the motion is tunneling through the barrier betweensites; U > 0 is the interaction energy, with interactions taken to be repulsive and on-site, and therefore short-range

interactions; and µ ≡ ∂E/∂N is the chemical potential or single particle energy, which controls the average filling.The extended Bose Hubbard model adds nearest-neighbor interactions; and often disorder is also considered throughyet another term; however, we neglect such terms here for the purposes of our discussion.

The boson Hubbard model is related to the quantum rotor model that we studied in Sec. IV. The rotor Hamiltonianwas invariant under O(N) rotation of the rotor orientation ni and the angular momentum Li. The Bose Hubbard

Hamiltonian Eq. (247) is invariant under a global U(1) ≡ O(2) phase transformation bi → bieiφ. The hopping term t

34

in Eq. (247) is similar to the J term in HR from Eq. (182). The quantum rotor coupled neighboring sites in such away that the rotor took on a particular orientation, breaking O(N) symmetry; so we can expect the Bose HubbardHamiltonian to do the same for phase symmetry. The competition between J and Jg in the quantum rotor model ledto two phases; can we expect two phases here as well, in the competition between t and U , with a quantum criticalpoint between?

There is one significant difference between the Bose-Hubbard and quantum rotor Hamiltonians. Consider the thirdterm in Eq. (247). The chemical potential in the Bose-Hubbard Hamiltonian appears analogous to the external fieldH in the quantum rotor Hamiltonian. But in the quantum rotor Hamiltonian we only examined linear response toH. This was because even infinitesimal H already broke the O(N) symmetry: mathematically, [L, HR] = 0 if and

only if H = 0, where L ≡∑i Li. But the Bose-Hubbard Hamiltonian always commutes with the analogous operator,

the total number, N ≡∑i ni, irrespective of the value of the chemical potential. This means that there is no obviousvalue of µ that we should look at based on symmetry, and we have to consider all possible values of the chemicalpotential. So immediately we see that the Bose-Hubbard Hamiltonian requires three energy scales, not just two.

To solve the Bose-Hubbard Hamiltonian, we begin with a mean field theory. Our solution method sometimes goesby the name of the Gutzwiller ansatz, and involves replacing off-site terms by their quantum averages:

HMF =∑

i

[

−Ψ∗bi − b†iΨ +U

2ni(

ni − 1)

− µni

]

, (248)

where Ψ ∈ C is a complex scalar which serves as a variational parameter; we have absorbed t into Ψ, without lossof generality. The terms with Ψ break the U(1) symmetry and do not conserve the number of particles; this termhas to be self-consistently determined, and allows for broken-symmetry phases. In this broken symmetry phase thesuperfluid stiffness we used to characterize the broken symmetry state of the quantum rotor will now become thesuperfluid density. We note that Eq. (248) assumes a uniform solution in space.

How can we solve Eq. (248)? Since there is no interaction or hopping between sites, Eq. (248) can be solved forarbitrary Ψ. The result is a product wavefunction between sites; that is, there is no entanglement between sites. Thenwe take the expectation value of HB in terms of this wavefunction. We subtract HMF from HB, take the average,and factorize. The result is

〈HB〉M

=E0

M=EMF(Ψ)

M− Zt〈b†〉〈b〉 + Ψ∗〈b〉 + 〈b†〉Ψ , (249)

where M is the number of lattice sites and Z is the number of nearest neighbors in the lattice, called the coordination

number. Then one can numerically minimize E0 over variations in Ψ, truncating the possible values of the on-siteboson number at some large value ni ≫ 1. The result is the mean field phase diagram, shown in Fig. 2.

In order to understand the different phases in Fig. 2, it is useful to consider limits of the parameter space. Fort = 0 the Hamiltonian Eq. (247) factorizes, and the mean field Eq. (248) is exact. The variational parameter Ψ = 0

and we only need to minimize the on-site interaction energy to minimize 〈HB〉. We encourage the reader to do thisfor her/him self. The result is

|ψ〉 =∏

i

|ni = n(µ/U)〉i , (250)

where

n =

0, µ/U < 01, 0 < µ/U < 12, 1 < µ/U < 2...

...n, n− 1 < µ/U < n

(251)

Each site has exactly the same integer number of bosons, with the particular integer controlled by the ratio of thechemical potential to the interaction energy. At integer values of µ/U a superposition of the adjacent number statesis allowed, but only for exactly t = 0. This is the vertical axis of Fig. (366).

Let us consider the perturbative effect of small nonzero t. As shown in Fig. 2, the regions of Ψ = 0 survive inlobes characterized by a given integer value of n(µ/U). One can show that there is a gap for t = 0; as a result, if weincrease t adiabatically, the system responds without any energy level crossings (avoided or unavoided). We observe

that Eq. (250) is an eigenstate of the total number operator N . The perturbation by the tunneling term of strength

35

t/U0

1

2

3

µ/U

SF

MI: n=1

MI: n=2

MI: n=3

-

-

-

n=2-ε

n=1+ε-

-

FIG. 2: A schematic plot of the phase diagram of the Bose-Hubbard model [2]. MI ≡ Mott Insulator and SF ≡ superfluid.Dashed lines show trajectories within the phase diagram of constant density.

t also commutes with N . Therefore, even though mean field theory becomes a poorer and poorer model for larger t,n remains the average number of bosons within the lobes even for t 6= 0, i.e.,

n = 〈ni〉 , (252)

for all i. This also means that there is always a gap within the lobes. The region inside of the lobes, which alwayshave a quantized value of the density and a gap, are called Mott insulators. Mott insulators are also incompressible

because the number of particles remains fixed under changes of the chemical potential,

∂〈N〉∂µ

= 0 . (253)

The variational mean field analysis can be used to show that the borders in the phase diagram of Fig. (2) correspondto second order quantum phase transitions. Then we can use the Landau theory of second order phase transitions [28],which says that near such a phase transition the energy can be expanded in even powers of the order parameter:

E0 = E00 + r|Ψ|2 + O(|Ψ|4) , (254)

and the phase boundary occurs when r changes sign. The value of r is determined from Eqs. (248) and (249) bysecond-order perturbation theory. The result is

r = χ(µ) [1 − Ztχ(µ)] , (255)

χ(µ) ≡ n+ 1

Un− µ+

n

µ− U(n− 1), (256)

µ ≡ µ

U, (257)

where n was defined in Eq. (251). The solution of r = 0 yields the phase boundaries sketched in Fig. (366).Outside the Mott insulating lobes, Ψ 6= 0. The order parameter Ψ varies continuously in response to changes in

the parameters of the Hamiltonian, and one finds a compressible state

∂〈N〉∂µ

6= 0 (258)

The U(1) phase symmetry is broken and the stiffness in response to twists in the order parameter is also nonzero, inanalogy to the quantum rotor. This is a superfluid phase, as can be determined more carefully from the considerationsof Sec. VB 2.

36

2. Second Mean Field Approach: Hubbard-Stratanovich Transformation

We would now like to go to a more precise description which includes fluctuations beyond what the mean fieldsolution provided. To do this, we use the Hubbard-Stratanovich transformation [29, 30], which introduces an auxiliarycomplex field Ψ(x, τ) in the continuum limit of the Bose-Hubbard theory. We again use field theoretic path integrationmethods which may be unfamiliar to some readers; in this case, simply try to follow along with the main points, notworrying about the mathematical details.

We begin with the partition function

ZB = Tre−HB/T (259)

in the standard imaginary-time coherent state path integral for canonical bosons,

ZB =

∫

Dbi(τ)Db†i (τ) exp

(

−∫ 1/τ

0

dτLB)

, (260)

LB =∑

i

(

b†idbidτ

− µb†ibi +U

2b†ib

†ibibi

)

− t∑

<i,j>

(

b†ibj + b†jbi)

, (261)

The auxiliary field Ψ(x, τ) is used to decouple the hopping term. We transform the partition function and Bose actionto

ZB =

∫

Dbi(τ)Db†i (τ)DΨi(τ)DΨ†i (τ) exp

(

−∫ 1/τ

0

dτL′B

)

, (262)

L′B =

∑

i

(

b†idbidτ

− µb†ibi +U

2b†ib

†ibibi − Ψib

†i − Ψ∗

i bi

)

+∑

i,j

Ψ∗i tijΨi , (263)

where tij is a matrix which is t for i and j nearest neighbor and zero otherwise. The equivalence of Eqs. (260)-(261)and (262)-(263) can be proven by performing the Gaussian integral and renormalizing Ψ.

We observe that Eq. (263) is invariant under time-dependent U(1) transformations:

bi → bieiφ(τ) ,

Ψi → Ψieiφ(τ) ,

µ → µ+ i∂φ

∂τ. (264)

This is a useful symmetry.

Next, we integrate out the fields bi and b†i from Eq. (263). This can be done exactly in powers of the auxiliaryfields Ψ and Ψ∗; the coefficients are products of Green’s functions of the bosonic annihilation and creation operators.This is analytically tractable because all the Ψ-independent part of Eq. (263) is a sum of single site Hamiltonians, forwhich we already determined that the product state Eq. (250) is a solution. All single site Green’s function can thenbe determined. We re-exponentiate the resulting series in powers of Ψ and Ψ∗ and expand each term in spatial andtemporal gradients of Ψ. Then the partition function becomes

ZB =

∫

DΨ(x, τ)DΨ∗(x, τ) exp

(

−VF0

T−∫ 1/T

0

dτ

∫

ddxLB)

, (265)

LB = K1Ψ∗ ∂Ψ

∂τ+K2

∣

∣

∣

∣

∂Ψ

∂τ

∣

∣

∣

∣

2

+K3|∇Ψ|2 + r|Ψ|2 +u

2|Ψ|4 + · · · , (266)

where V ≡ Mad is the lattice volume, M is the number of sites, and F0 is the free energy density of a system ofdecoupled sites. By the arguments we made in Sec. (V B1), the derivative of F0 with respect to the chemical potentialyields the density of the Mott insulator,

−∂F0

∂µ=n(µ/U)

ad. (267)

37

All parameters in Eq. (266) can be expressed in terms of the Bose Hubbard Hamiltonian energy scales µ, U , and t.The most important parameter is r which can be shown to be

rad =1

Zt− χ(µ/U) , (268)

where χ is defined in Eq. (256). Interestingly, r changes sign for the same parameter values as r, so that the meanfield critical boundary (critical point) between the two phases is the same whether using the Gutzwiller or Hubbard-Stratanovich approaches.

The coefficient of the first-order time derivative in Eq. (266), K1, is important. It can be fixed by requiring thatEq. (266) have the symmetry Eq. (264) for small φ. Then it follows that

K1 = − ∂r

∂µ. (269)

We observe that K1 vanishes when r does not depend on µ; this is the same as requiring that the phase boundary inFig. 2 have a vertical tangent, as occurs at the tips of the Mott lobes. It turns out that setting K1 = 0 in Eq. (266)yields the quantum rotor action for N = 2. Thus, right at the top of the Mott lobes, the Mott insulator to superfluidtransition is in the same universality class as the O(2) quantum rotor. Away from the tip, where K1 6= 0, we find anew, different universality class. Thus the Bose-Hubbard Hamiltonian contains two universality classes.

In fact, the Mott lobe tips turn out to be important in different spatial dimensions as well. One can also determine aphase diagram for the Bose-Hubbard Hamiltonian without a mean field approximation of any kind, via the numericalmethod of Sec. IX, at least in one spatial dimension; quantum Monte Carlo presents another solution method whichcan be used in higher dimensions. The result is close to the mean field prediction in three dimensions; in 2D one findsa slight cusp at the outer edges of the lobes; and in 1D the lobes take a very different shape, with pronounced cusps.

Finally, we can make a connection between universality classes and the boson density across the quantum phasetransition. We note that

〈ni〉 = 〈b†i bi〉 = −ad ∂F0

∂µ− ad

∂FB∂µ

= n(µ/U) − ad∂FB∂µ

, (270)

where FB is the free energy from the functional integral over Ψ in Eq. (266). In mean field theory, for r > 0, Ψ = 0and FB = 0; so

〈b†i bi〉 = n(µ/U) for r > 0 . (271)

Thus the system is Mott insulating. In contrast, for r < 0 we expect to find the opposite phase according to Landautheory. From minimization of LB in Eq. (266), Ψ = (−r/u)1/2. Then computation of the free energy leads to

〈b†i bi〉 = n(µ/U) + ad∂

∂µ

(

r2

2u

)

≃ n(µ/U) +adr

u

∂r

∂µ. (272)

In the approximation, we neglect the derivative of u because it is less singular as u→ 0. Thus at the tip of the Mottlobe where K1 = 0, the leading correction to the density of the superfluid phase vanishes, and it remains at the samevalue as it had throughout the Mott lobe. In contrast, for K1 6= 0 the transition is always accompanied by a densitychange. This pins down the concept of two universality classes in this model in terms of an experimentally observablequantity.

C. Fermi-Bose Hubbard Hamiltonian

We know that the constituents of matter are fermions: protons, neutrons, and electrons. Yet we find many effectivebosons in Nature including quasiparticles such as phonons and bosonic atoms such as 87Rb. How is it that fermionscome together to make bosons? In particular, how does this happen in the context of a quantum lattice problem,and what quantum phases might we find under these circumstances? This question in many body theory becamewell-formulated starting in the 80’s; in the context of superconductivity it is called the “BCS-BEC crossover” problem.

One proposal [3, 31–34] for a Fermi-Bose Hubbard Hamiltonian (FBHH) is in terms of a two-channel model,

38

Internuclear distanceIn

tera

tom

ic p

oten

tial

open channel

closed channel

ν

(b)

(a)

FIG. 3: Outer figure: pairs of fermionic atoms in the open channel are coherently transferred into a closed channel, bosonicstate via a Feshbach resonance. Inset: second order degenerate perturbation theory in the limit Jf ≪ Vf leads to two hoppingevents on the lattice, (a) pair hopping, and (b) a single fermion hopping to an adjacent site and back.

motivated by a Feshbach resonance in an ultracold Fermi gas. Consider the FBHH in the grand canonical ensemble,

H = Hf +Hb +Hfb , (273)

Hb ≡ −Jb∑

〈i,j〉(b†ibj + bib

†j)

+1

2Vb∑

i

nbi (nbi − 1) − µb

∑

i

nbi , (274)

Hf ≡ −Jf∑

〈i,j〉,s,m,m′

(f †ismfjsm′ + fismf

†jsm′)

− 1

2Vf

∑

i,m,m′

nfi↑mnfi↓m′ −

∑

i,s,m

(µf − Em)nfism , (275)

Hfb ≡ g∑

i

(b†ifi↑fi↓ + bif†i↓f

†i↑) +

Vfb2

∑

i,s,m

nbinfism . (276)

Equations (274)-(275) are the usual repulsive Bose-Hubbard and multi-band attractive Fermi-Hubbard Hamiltoniansfor a uniform lattice and Eq. (276) is the fermion-boson coupling. The symbol 〈i, j〉 denotes nearest neighbors, whilethe indices s ∈ ↑, ↓ and m denote the spin state and band number. The hopping or tunnelling strengths Jf,b and theon-site interaction strengths Vf,b are taken as real and positive definite. The band gap energy of the mth band is Em.The strength g of the interconversion term and Vfb of the density coupling may have either sign. The creation andannihilation operators f †, f and b†, b satisfy the usual commutation relations for fermions and bosons, respectively.

The number operators are defined as nbi = b†ibi, nfism = f †

ismfism. In order to match the physical context of quantumdegenerate gases in chemical equilibrium, we require

µb = 2µf + ~ν , (277)

where ν is the detuning associated with a Feshbach resonance and we set ~ = 1. The conserved quantity

n ≡ 2∑

i nbi +

∑

i,s,m nfism (278)

is the total number of fermions. Eliminating µb by substituting Eq. (277) into Eqs. (273)-(276), one finds that µfmultiplies n. One can thus take µf as the chemical potential of the coupled system, while ν determines the relativenumber of bosons and fermions.

The FBHH of Eqs. (273)-(276) models a pseudo-spin-1/2 system of fermions with s-wave interactions, as in exper-iments [35–37]. In practice, the index s ∈ ↑, ↓ represents two hyperfine states in the level structure of an effectivelyfermionic alkali atom, such as 40K or 6Li, scattering near threshold in an open channel. The bosonic field representsa bound closed-channel molecular state, 6Li2 or 40K2, which is coupled to the fermionic field via a resonance with

39

an unbound open-channel atomic state, called a Feshbach resonance. A schematic is shown in Fig. 3. Note that Vf,band g are not functions of ν. Methods for calculating the parameters Vf , Vb, etc. in Eqs. (274)-(276) from few-bodyatomic physics have been described in detail elsewhere [33]. Another important assumption is that the pairing offermions into bosons occurs on-site. This is physically reasonable for present experiments [38].

We consider the limit in which Jf ≪ Vf , which corresponds to a strongly confining lattice [39]. Since the latticeheight is proportional to the intensity of the lasers creating the standing wave, this is straightforward to obtain.Because the on-site interactions are attractive and s-wave, and the hopping is taken perturbatively, the fermions formspin-up/spin-down pairs. We also restrict them to be in the lowest band. This is the typical experimental case inthree dimensions, where 105 to 106 fermions are distributed among 1003 sites. Thus m = 1 and n ∈ [0, 2], i.e., thereare from zero to two fermions, or zero to one fermi pair, per site. Then second order degenerate perturbation theorymaps Hf onto a new spin-1/2 system (a quantum XXZ model) [40]:

H ′f = −J ′

f

∑

〈i,j〉(τ+i τ

−j − ninj) − µ′

f

∑

i

ni , (279)

where J ′f ≡ 8J2

f /Vf and µ′f ≡ 2µf + Vf/2. The operator τ+

i ≡ (τ−i )† ≡ f †i↓f

†i↑ is a pair creation/annihilation operator

and ni ≡ 12 (nfi↑ + nfi↓ − 1). The perturbative treatment results in two hopping-type events, as is sketched in the inset

of Fig. 3: the τ+i τ

−j term corresponds to pair hopping, while the ninj term corresponds to a single fermion hopping

to an adjacent site and hopping back.

These operators obey the commutation relation [ταi , τβj ] = 2iτγi ǫ

αβγδij , where α, β, γ ∈ x, y, z, τ±i ≡ τxi ± iτyi ,

and τzi ≡ ni. Thus, despite the fact that τ±i is a creation/annihilation operator for fermion pairs, the τ operatorsobey the Pauli spin commutation relations, not the bosonic commutation relations. This is one reason why theattractive Fermi-Hubbard model does not map simply onto the repulsive Bose-Hubbard model, even in the limit ofstrong interactions. A second reason is that in order to achieve such a mapping, a sum over many bands is required,since the internal energy of bosons composed of two fermions is much greater than the band spacing. In contrast, theFBHH is asymptotically able to represent both the attractive Fermi-Hubbard and repulsive Bose-Hubbard models ina simple way. It is therefore a good candidate for the study of the BCS-BEC crossover.

In general, a paired Fermi Hubbard Hamiltonian can act on all number states of the fermions. However, as inEq. (279) we consider only n ∈ [0, 2], the Hilbert space on which it operates is restricted to two paired-number states.Thus H ′

f is equivalent to the Heisenberg spin Hamiltonian, or a magnet,

Hspin = −∑i,j JijSi · Sj − h ·∑i Si , (280)

where µ′f plays the role of the magnetic field hz. For T = 0, one therefore expects coherent paramagnetic and either

ferro- or anti-ferromagnetic phases. The former correspond to the superfluid phase, while the latter are Mott [41] andcharge-density wave (checkerboard) phases. Similarly, the restriction of the Hilbert space on which Hb operates totwo number states leads to an isotropic Heisenberg spin Hamiltonian (the quantum XX model [1]). We formulate thetwo-state approximation for the coupled model as superposition states of the form |ψ〉 =

∏

j |ψ〉j , where

|ψ〉j ≡ |0〉bj ⊗ |0〉fj cos θj + sin(θj) eiφj

× (|1〉bj ⊗ |0〉fj cosχj + |0〉bj ⊗ |1〉fj sinχj eiαj ) . (281)

The superscripts b and f refer to Fock states of bosons and fermi pairs on the jth site.The two state approximation is useful in determining the Mott-superfluid borders in the phase diagram. The

Mott state is a single number state, while the lowest order approximation of a superfluid is a superposition of twonumber states. Therefore, Mott states occur in Eq. (281) for θ ∈ 0, π/2, π. The mixing angle χ is determined bythe detuning ν in Eq. (277). To determine which phase is energetically favorable one evaluates Egs ≡ 〈ψ|H |ψ〉. Inaddition, we make the uniform approximation θj = θ , φj = φ, χj = χ , αj = α. Then φ does not appear in theground state energy, while α can only change the sign of g. Setting g′ = min[g exp(iα)], neither φ nor α need beconsidered to obtain the phase diagram. Note that the uniform approximation leaves out a range of excited manybody states. However, it does obtain two solutions vital to understanding the crossover, one of which is the groundstate, as explained below and in Fig. 5. An important point is that the ground state is either coherent paramagnetic(superfluid) or ferromagnetic (Mott). It can be proven that it is not antiferromagnetic (charge-density wave), eitherby setting the angles to differ by π/2 on each site, or by making a spin rotation in the Hamiltonian [42].

The Mott-superfluid borders are obtained as follows [43]. The ground state energy is expanded around the Mottangles θ ∈ 0, π/2, π. The zeroth order term gives the energy. The first order term is zero, showing that the Mottstate is always an extremum. The sign of the second order term determines whether the Mott state is a maximum

40

0 0.05 0.1 0.15 0.2J

f’ / V

f

-2

0

2

4

6

µ f / V

f

MI n=0

MI n=2SF

0 0.05 0.1 0.15 0.2J

f’ / V

f

-2

0

2

4

6

µ f / V

f MI n=0

MI n=2SF

0 0.05 0.1 0.15 0.2J

f’ / V

f

-2

0

2

4

6

µ f / V

fMI n=0

MI n=2SF

0 0.05 0.1 0.15 0.2J

f’ / V

f

-2

0

2

4

6

µ f / V

f

MI n=0

MI n=2SF

(a) (b)

(c) (d)

FIG. 4: (color online) Shown is the phase diagram for detunings (a) ν/Vf = −10, (b) ν/Vf = −1, (c) ν/Vf = 1/2, (d)ν/Vf = 10. The blue solid curves show the Mott-superfluid borders, while the red dashed curves show alternate extrema whichare maxima (see Fig. 5). SF ≡ superfluid, MI ≡ Mott insulator, n ≡ fermion filling factor.

or a minimum. Setting this equal to zero, one obtains the Mott-superfluid borders. One must also extremize in themixing angle χ and determine whether or not it is a maximum. Thus there are three conditions:

∂2Egs/∂θ2 = 0 , (282)

∂Egs/∂χ = 0 , (283)

∂2Egs/∂χ2 > 0 . (284)

Using conditions (282)-(283) to eliminate χ and Eq. (277) to eliminate µb, one finds a quartic equation in µf . Thecoefficients are functions of J ′

f , Jb, Vf , Vb, |g′| = |g|, and ν. We set Vfb = 0 in order to focus on the effect of the

fermion-boson conversion term in Eq. (276). The solution to the quartic equation, though lengthy, can be written inclosed analytic form. It is best understood when evaluated in limits of the parameters and for particular values ofthem.

First consider the case ν → ±∞. We assume a bipartide lattice with Z the number of nearest neighbors. Thenχ ∈ 0, π/2 and one obtains the Mott borders

µf/Vf = −1/4 + (Z/2)(1 − 2σf )J′f/Vf , (285)

µb/Vb = −2σbZJb/Vb , (286)

where σf ≡ ±1 gives the vacuum/one-fermi-pair and σb ≡ ±1 the vacuum/one-boson Mott states. Equations (285)-(286) correspond to the solutions one finds for g = 0 in the two-state approximation. For ν → +∞, condition (c)shows that Eq. (285) is a minimum and Eq. (286) is a maximum. For ν → −∞, the inverse is the case. Thus theBose Hubbard and paired-Fermi Hubbard limits are obtained naturally from the ansatz of Eq. (281) in the limits oflarge negative and positive detuning. The FBHH we have proposed therefore correctly obtains the endpoints of theBCS-BEC crossover on a lattice.

Next consider the case of the physically reasonable parameter set Vb = Vf , Jb = J ′f , with the scaling chosen such

that Vf = 1. The quartic equation has four roots. Two are complex and therefore physically extraneous. The othertwo represent an energy minimum and an energy maximum. There is no saddle point. The phase diagram is shownin Fig. 4 for ν = −10,−1, 1/2, 10 and g = 1. The results are qualitatively the same for all g 6= 0. The point ν = 1/2is the actual crossover in our model, i.e., the point at which the Mott borders become degenerate. To illustrate this,in Fig. 5(a) is shown the mixing angle χ as a function of ν. Note the appropriate ν → ±∞ limits. In Fig. 5(b) areshown the y-intercepts of the Mott phases from the phase diagrams of Fig. 4 as a function of ν. These go through anavoided crossing [44] at ν = 1/2. Smaller values of |g| cause the avoided crossing to become narrower. Similarly, thewidth of χ(ν) in Fig. 5(a) is proportional to |g|.

41

-10 -5 0 5 10ν/V

f

0

0.1

0.2

0.3

0.4

0.5

χ/π

-10 -5 0 5 10ν/V

f

-6

-4

-2

0

2

4

6µ f /

Vf (J

f’ =

0)

(a)

(b)

FIG. 5: (color online) (a) The mixing angle χ as a function of the detuning ν/Vf . (b) The y-intercepts in the phase diagramof Fig. 4 go through an avoided crossing as a function of the detuning. Blue solid curves: energy minima; red dashed curves:energy maxima.

D. Molecular Hubbard Hamiltonian

What about driven systems? So far we have only considered H 6= H(t). What happens when the parameters inthe Hamiltonian become a function of time? Is the concept of a quantum phase transition still relevant? What do wemean by a phase diagram for a dynamical system?

One example of such a Hamiltonian is the Molecular Hubbard Hamiltonian [5]:

H = −∑

JJ′M

tJJ′M

∑

〈i,i′〉

(

a†i′,J′M aiJM + h.c.)

+∑

JM

EJM∑

i

niJM − π sin (ωt)∑

JM

ΩJM∑

i

(

a†iJ,M aiJ+1,M + h.c.)

+1

2

∑

J1, J′1, J2, J′

2M, M′

UJ1, J′

1, J2, J′2

M, M′

dd

∑

〈i,i′〉a†iJ1M

aiJ′1M a

†i′J2M ′ ai′J′

2M ′ . (287)

In Eq. (287) the hopping t depends on rotational modes |JM〉 due to use of a dressed (rotational plus DC electricfield) basis, and the dependence of the optical lattice potential energy on the molecular polarizability has been takeninto account. The other three terms in Eq. (287), in order, represent the DC electric field, the AC electric drivingfield, and the dipole-dipole interactions, which have been taken as hard core. Equation (287) can be viewed as amodified, multiband, driven, extended Bose Hubbard Hamiltonian; with no driving and only a single rotational stateoccupied, it is exactly the extended Bose Hubbard Hamiltonian [45]. We present Eq. (287) to give one example of howmuch more complicated MHHs are compared to, for example, the Bose Hubbard model of Eq. (247). Thus advancedsimulation techniques can be required [46] in order to understand experiments on these systems and yield new insightinto complex quantum dynamics and emergent properties. In Sec. IX we will discuss some of these techniques.

Without going into too much detail, the length and energy scales of the problem are as shown in Table I. The basicphysical system modeled by Eq. (287) is that of heteronuclear polar molecules in an optical lattice. It turns out thatEq. (287) gives rise to two emergent time scales, one for spatial entanglement, and the other for self-damping towards

42

Term Length scale Energy scale

Rotation internuclear distance ∼ 1 A B ∼ 60~ GHz ≈ 2cm−1

DC field N/A, uniform dEDC ∼ 120~ GHz ≈ 4cm−1

AC field 2πc/ω ∼ 1cm ~ω ∼ 30~ GHz ≈ 1cm−1

Kinetic l(00)ho,x ∼ 100nm ER ∼ 1.46~ kHz

Tunneling Lattice spacing∼ 1µm tJ′JM ∼ 100~ Hz

Resonant Dipole-Dipole energy comparable to B∣

∣

∣〈E ; 00|d|E ; 00〉

∣

∣

∣

2

/ (1µm)3 ∼ 1.2~ kHz

at rB ≃ 348 Bohr radii for nearest neighbors

TABLE I: Comparison of energy and length scales for the Molecular Hubbard Hamiltonian of Eq. (287).

a supersolid which is a superposition of rotating and non-rotating molecules, as we will show. In order to analyzethese time scales, we apply the following quantum measures.

1. Quantum Measures

We use a suite of quantum measures to characterize the reduced MHH, Eq. (292) below. The few-body measures

we use are 〈nJi 〉, the number in the J th rotational state on the ith site, E ≡ 〈H〉, the expectation of the energy,and 1

L〈nJ 〉, the average number in the J th rotational state per site (L is the number of lattice sites). The latter is aJ-dependent filling factor. The many body measures we use include the density-density correlation between rotationalmodes J1 and J2 evaluated at the middle site

g(J1J2)2

(

⌊L2⌋, i)

≡ 〈n(J1)

⌊L2⌋n

(J2)i 〉 − 〈n(J1)

⌊L2⌋〉〈n

(J2)i 〉, (288)

where ⌊q⌋ is the floor function, defined as the greatest integer less than or equal to q. As an entanglement measurewe use the Meyer Q-measure [47–49]

Q ≡ dd−1

[

1 − 1L

∑Lk=1 Tr

(

ρ(k))2]

, (289)

where ρ(k) is the single-site density matrix obtained by tracing over all but the kth lattice site, and the factor outside ofthe bracket is a normalization factor (d is the on-site dimension). This gives an average measure of the entanglementof a single site with the rest of the system. The Q-measure can also be interpreted as the average local impurity(recall that the Tr(ρ2) = 1 if and only if ρ is a pure state).

To determine what measures we can use to ascertain the static phases of our model we reason by analogy with theextended Bose-Hubbard Hamiltonian where we know that the possible static phases are charge density wave, super-fluid, supersolid, and Bose metal [50]. The charge density wave is an insulating phase appearing at half integer fillingswhich has a wavelength of two sites. Like the Mott insulating phase, it has an excitation gap and is incompressible.While the extended Bose-Hubbard Hamiltonian has only one charge density wave phase due to the presence of onlyone species, the MHH has the possibility of admitting several charge density wave phases due to the presence ofmultiple rotational states. As such, we define the structure factor

S(J1J2)π = 1

N

∑

ij (−1)|i−j| 〈n(J1)

i n(J2)j 〉 , (290)

where N is the total number of molecules. We recognize this object as the spatial Fourier transform of the equal-timedensity-density correlation function between rotational states J1 and J2, evaluated at the edge of the Brillouin zone.This measure is of experimental interest because it is proportional to the intensity in many scattering experiments,e.g. neutron scattering. Crystalline order between rotational states J1 and J2 is characterized by a nonzero structure

factor S(J1J2)π . The charge density wave is the phase with crystalline order but no off-diagonal long-range order as

quantified by the superfluid stiffness of rotational state J

ρ(J)s = lim

φ→0L∂2E(J) (φ, L)

∂φ2(291)

43

(note that ρs bears no relation to the density matrix ρ). If both the structure factor and the superfluid stiffness arenonzero, the phase is called supersolid. If both the structure factor and the superfluid stiffness are zero, the phase iscalled Bose Metal. Finally, if the structure factor is zero and the superfluid stiffness is nonzero, the phase is superfluid.In one dimensional systems with short range interactions the structure factor is zero in the thermodynamic limit andthe entire superfluid phase is critical, thus there are no order parameters [50]. Superfluidity is instead signaled by adiverging correlation length and solid order by slow power law decay of the density-density correlator.

2. Case Study: Hard Core Bosonic Molecules at Half Filling

In the following, we consider a particular case of Eq. (287) for dynamical study. We choose the hard core case, whichcan occur naturally due to strong on-site dipole-dipole interactions, and half filling, which is an interesting point in anumber of models, including the repulsive Fermi-Hubbard Hamiltonian and the extended Bose-Hubbard Hamiltoniandiscussed in Sec. VD1. For example, in the latter case, the charge-density-wave phase requires a minimum of half-filling [50].

If we assume that our system begins in its ground state (J = 0, M = 0) we need only include states which have adipole coupling to this state. For z-polarized DC and AC fields, this means we only consider M = 0 states, yieldingthe reduced Hamiltonian

H = −∑JJ′ tJJ′

∑

〈i,i′〉

(

a†i′,J′ aiJ + h.c.)

+∑

J EJ∑

i niJ − π sin (ωt)∑

J ΩJ∑

i

(

a†iJ aiJ+1 + h.c.)

+ 12

∑

J1,J′1,J2,J′

2UJ1,J

′1,J2,J

′2

dd

∑

〈i,i′〉 a†iJ1aiJ′

1a†i′J2

ai′J′2. (292)

This is the specific case of the MHH that we study using TEBD.A matter of practical concern, as apparent in Table I, is the large disparity between the timescales of the first three

(Rotational, DC, and AC) and the last three (kinetic, tunneling, and Dipole-Dipole) terms. The accumulation oferror resulting from truncating the Hilbert space at each TEBD timestep causes the algorithm to eventually fail aftera certain “runaway time,” making studies over long times intractable [51]. This invites a multiscale approach in thefuture [52, 53]. In our current numerics we artificially increase the recoil energy and dipole-dipole potential to be ofthe order of the rotational constant in order to study Eq. (292) using TEBD. In particular, we take

UJ1,J

′1,J2,J

′2

dd = 10Bd2 〈E ; J ′

1|d|E ; J1〉〈E ; J ′2|d|E ; J2〉 , (293)

tJ = 10B[

η(

1 + 2∆αα

J(J+1)(2J+1)(2J+3)

)]1.051

× exp

[

−2.121

√

η(

1 + 2∆αα

J(J+1)(2J+1)(2J+3)

)

]

, (294)

where the dimensionless variable η becomes an ersatz “lattice height.” To see the scaling more explicitly, we comparethe above with the actual expressions for the MHH parameters

UJ1,J

′1,J2,J

′2

dd = 8λ3 〈E ; J ′

1|d|E ; J1〉〈E ; J ′2|d|E ; J2〉 (295)

=(

2mERd4/3

~2π2

)32 〈E ; J ′

1|d|E ; J1〉〈E ; J ′2|d|E ; J2〉/d2 , (296)

tJM ≈ 1.397ER

(

|Eopt|2α3ER

[

1 + 2∆αα

J(J+1)−3M2

(2J−1)(2J+3)

])1.051

(297)

× exp

(

−2.121

√

|Eopt|2α3ER

[

1 + 2∆αα

J(J+1)−3M2

(2J−1)(2J+3)

]

)

. (298)

If we now scale ER to be 10B/1.397 and set d such that[

2mERd4/3/

(

~2π2)]

32 = 10B for this ER, we recover

Eqs. (293) and (294) provided we make the definition

η ≡ − |Eopt (x)|2 α/ (3ER) = V(JM)x α/

(

3ERα(t)JM

)

. (299)

Since this dimensionless parameter plays the same role as the quasi-1D lattice height scaled to the recoil energy didin the actual MHH, we refer to it as the lattice height. For the polarizability tensor, we choose ∆α/α = 165.8/237,

44

L η τB/~ Asymp. S.E. τQB/~ Asymp. S.E.

9 1 414.04 0.72% 398.4 0.51%

9 2 224.32 1.79% 149.9 1.36%

9 3 117.5 1.86% 126.7 1.03%

9 10 613.00 1.07% 1079.66 14.09%

10 1 259.96 0.76% 240 0.6454%

10 4 140.70 1.19% 72.04 0.60%

10 10 526.21 0.88% 396.46 1.018%

21 1 756.18 3.13% 110.68 0.96%

21 5 177.53 1.62% 75.18 0.902%

21 10 716.21 2.96% 244.09 2.82%

TABLE II: Emergent time scales τ and τQ and their fit asymptotic standard errors for various lattice heights and system sizes.

corresponding to LiCs. This rescaling does not change the qualitative static and dynamical features of Eq. (292);it only makes Eq. (292) treatable directly by TEBD, without multiscale methods. In the future, we plan to applymultiscale methods to determine the emergent time scales for experimentally relevant parameters.

Our main focus at present is on the dynamics of the MHH. In the following numerical study, we explore dynamicsas a function of the physical characteristics of the lattice, namely, number of sites L and effective lattice height η.Specifically, we study L = 9, 10, and 21 lattice sites with N=4, 5, and 10 molecules, respectively, and η rangingfrom 1 to 10. We fix the dipole-dipole term as in Eq. (293), and fix the DC field parameter to be βDC = 1.9. WhileβDC = 1.9 may not correspond to a physically realizable situation, its exploration provides insight into the MHH.

The Rabi oscillations between the J = 0 and the J = 1 states damp out exponentially in the rotational timetr ≡ Bt/~ as

〈n0〉 = a0 − b0 e−tr/τ cos (c0tr) , (300)

〈n1〉 = a1 − b1 e−tr/τ cos (c1tr) , (301)

with some characteristic time scale τ , as seen in Fig. 6. We note that an exponential fit has a lower reduced chi-squaredthan a power-law, or algebraic fit. We also tried fit functions where the oscillations do not decay to zero, but ratherpersist with some asymptotic nonzero amplitude. We find that the fit functions Eqs. (300) and (301) above fit thedata better as quantified by the convergence properties of the algorithms used.

The time scale τ also describes the decay of physically measurable quantities, for example the structure factorsas defined in Eq. (290) and illustrated in Fig. 7. We show the emergent time scale τ for various lattice heights andsystems sizes in Table II.

Examining Fig. 6, one observes that the driven system approaches a dynamical equilibrium that is a mixture ofrotational levels. The time scale with which the system relaxes to this equilibrium, τ , cannot be determined fromthe single-molecule physics, and so we refer to τ as an emergent time scale. For the low lattice height η = 1, thepopulations of the first two rotational states appear to oscillate around and asymptotically converge to roughly quarterfilling, with J = 1 being lower due to contributing to population of J = 2 via an off-resonant AC coupling (Fig. 6(a)).For η = 5, the asymptotic equilibrium is an uneven mixture of rotational states that favors occupation of the J = 0state (Fig. 8(a)), and the emergent time scale for reaching this equilibrium is shorter than it was for η = 1 by roughlya factor of four. As the lattice height is then increased to η = 10, the populations return to the trend of η = 1, againconverging to quarter filling with a time scale comparable to that of η = 1 (Fig. 8(c)). This illustrates the fact thatthe emergent time scale τ is not, in general, a monotonic function of the parameters of the lattice.

While the dynamics of the site-averaged rotational state populations are superficially similar for η = 1 and η = 10,the underlying physics is not identical, as can be seen by comparing Figs. 6(f), 8(b), and 8(d). These figures displaythe squared modulus of the Fourier transform of the site-averaged number in the J = 0 state. The only significantfrequency observed for η = 1 is the Rabi frequency Ω ∼ 0.064B/~. In contrast, the η = 5 case has numerous othercharacteristic frequencies. As we raise the lattice height to η = 10, the frequencies that arose for η = 5 remain, eventhough the overall visual trend of the site-averaged number reflects that of the single-frequency η = 1 behavior. Whilewe do not explicitly see the new frequencies in the site-averaged number, we do see them in the structure factors. An

45

0 20 40 60 80 100 120 140 1600

0.1

0.2

0.3

0.4

0.5

Bt/h

〈nJM〉/L

L = 9, η = 1, βDC = 1.9

〈n00〉/L〈n10〉/L〈n20〉/L

(a)Site-averaged population vs. rotational time for 9 sites. Note thegeneral theme; a gradual decrease (increase) of the maxima

(minima) of oscillations.

0 0.5 1 1.5 210

−10

10−8

10−6

10−4

10−2

100

hω/B

|F[〈n

00〉]|2

L = 9, η = 1, βDC = 1.9

(b)Squared modulus of Fourier transform ofsite-averaged J = 0 population vs. rotationally scaledfrequency for L = 9 sites. The arrow denotes the Rabi

frequency Ω00.

0 20 40 60 80 100 120 140 1600

0.1

0.2

0.3

0.4

0.5

Bt/h

〈nJM〉/L

L = 10, η = 1, βDC = 1.9

〈n00〉/L〈n10〉/L〈n20〉/L

(c)Site-averaged population vs. rotational time for 10 sites. Notethat there is no significant difference between an odd and even

number of sites.

0 0.5 1 1.5 210

−10

10−8

10−6

10−4

10−2

100

hω/B

|F[〈n

00〉]|2

L = 10, η = 1, βDC = 1.9

(d)Squared modulus of Fourier transform ofsite-averaged J = 0 population vs. rotationally scaled

frequency for L = 10 sites.

0 20 40 60 80 100 120 140 1600

0.1

0.2

0.3

0.4

0.5

Bt/h

〈nJM〉/L

L = 21, η = 1, βDC = 1.9

〈n00〉/L〈n10〉/L〈n20〉/L

(e)Site-averaged population vs. rotational time for 21 sites. Notethat there is no significant difference between this and the smaller

system sizes.

0 0.5 1 1.5 210

−10

10−8

10−6

10−4

10−2

100

hω/B

|F[〈n

00〉]|2

L = 21, η = 1, βDC = 1.9

(f)Squared modulus of Fourier transform ofsite-averaged J = 0 population vs. rotationally scaled

frequency for L = 21 sites.

FIG. 6: Dependence of site-averaged number on lattice size L. For this set of parameters, the site-averaged J = 0 and J = 1populations appear to asymptotically approach quarter filling. The J = 2 mode is populated slightly by off resonant ACcouplings. The peak near the left side of the Fourier transform plots is the Rabi frequency Ω00, denoted by an arrow.

46

0 20 40 60 80 100 120 140 1600

0.2

0.4

0.6

0.8

1

1.2

Bt/h

S(J

1J

2)

π

L = 9, η = 1, βDC = 1.9

S(00)π

S(01)π

S(11)π

(a)Structure factors vs. rotational time for 9 sites. Note the similarasymptotic behavior to the populations in Fig. 6(a).

0 0.5 1 1.5 210

−15

10−10

10−5

100

hω/B

|F[

S(00)

π

]

|2

L = 9, η = 1, βDC = 1.9

(b)Squared modulus of Fourier transform of S(00)π

vs. rotationally scaled frequency for L = 9 sites. Notethe similarity with Fig. 6(b) above.

0 20 40 60 80 100 120 140 1600

0.2

0.4

0.6

0.8

1

1.2

Bt/h

S(J

1J

2)

π

L = 10, η = 1, βDC = 1.9

S(00)π

S(01)π

S(11)π

(c)Structure factors vs. rotational time for 10 sites. There is no

significant difference in the S(00)π and S

(11)π between even and odd

L. For the difference in S(01)π , see Fig. 7(f).

0 0.5 1 1.5 210

−15

10−10

10−5

100

hω/B

|F[

S(01)

π

]

|2

L = 9, η = 1, βDC = 1.9

(d)Squared modulus of Fourier transform of S(10)π

vs. rotationally scaled frequency for L = 9 sites. Notethe absence of the Rabi frequency.

0 20 40 60 80 100 120 140 1600

0.2

0.4

0.6

0.8

1

1.2

Bt/h

S(J

1J

2)

π

L = 21, η = 1, βDC = 1.9

S(00)π

S(01)π

S(11)π

(e)Structure factors vs. rotational time for 21 sites. Note the lack ofsignificant difference with the smaller odd system size.

0 50 100 150−0.03

−0.025

−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

Bt/h

S(01)

π

η = 1, βDC = 1.9

L = 9, 21L = 10

(f)Comparison of the S(01)π correlation structure factor

for odd and even numbers of sites. Note that the evensite (exactly half filling) structure factor grows fasterand larger than the odd site (slightly less than half

filling) structure factor.

FIG. 7: Dependence of structure factors within and between rotational states J on the number of lattice sites. We do notconsider the off-resonant J = 2 and higher rotational states because they have a very small occupation; J = 2 is shown explicitlyin Fig. 6.

47

0 50 100 1500

0.1

0.2

0.3

0.4

0.5

0.6

Bt/h

〈nJM〉/L

L = 21, η = 5, βDC = 1.9

〈n00〉/L〈n10〉/L〈n20〉/L

(a)Site-averaged population vs. rotational time for 21 siteswith η = 5. Note that the J = 0 and J = 1 states now

appear to converge to different fillings.

0 0.5 1 1.5 210

−15

10−10

10−5

100

hω/B

|F[〈n

00〉]|2

L = 21, η = 5, βDC = 1.9

(b)Squared modulus of Fourier transform of 〈n00〉vs. rotationally scaled frequency for L = 21 sites andη = 5. Note the presence of several new frequencies

not observed in the η = 1 case (Fig. 6(f)). Inparticular, Ω00, 2Ω00, and 3Ω00, are denoted by

arrows.

0 50 100 1500

0.1

0.2

0.3

0.4

0.5

0.6

Bt/h

〈nJM〉/L

L = 21, η = 10, βDC = 1.9

〈n00〉/L〈n10〉/L〈n20〉/L

(c)Site-averaged population vs. rotational time for 21 siteswith η = 10. Note the similarity to the η = 1 case (Fig. 6(e))

and the difference from the η = 5 case(Fig. 8(a))–theasymptotic behavior is not a monotonic function of the

lattice height.

0 0.5 1 1.5 210

−15

10−10

10−5

100

hω/B

|F[〈n

00〉]|2

L = 21, η = 10, βDC = 1.9

(d)Squared modulus of Fourier transform of 〈n00〉vs. rotationally scaled frequency for L = 21 sites and

η = 10. Note that the frequencies that emerged duringη = 5 have persisted.

FIG. 8: Dependence of the asymptotic behavior of rotational state populations on the lattice height η.

48

0 50 100 1500

0.2

0.4

0.6

0.8

1

1.2

Bt/h

S(J1J 2

)π

L = 21, η = 5, βDC = 1.9

S(00)π

S(01)π

S(11)π

(a)Structure factors vs. rotational time for 21 sites withη = 5.

0 50 100 150−0.04

−0.02

0

0.02

0.04

0.06

Bt/h

S(01)

π

L = 21, βDC = 1.9

η = 5η = 10

(b)Correlation structure factor S(01)π vs. rotational

time for 21 sites with η = 5, 10.

0 50 100 1500

0.2

0.4

0.6

0.8

1

1.2

Bt/h

S(J1J 2

)π

L = 21, η = 10, βDC = 1.9

S(00)π

S(01)π

S(11)π

(c)Structure factors vs. rotational time for 21 sites with

η = 10. Note the similarity of S(00)π and S

(11)π to the η = 1

case (Fig. 7(e)). Note also that S(01)π is now nonzero, and is

periodic with the Rabi frequency Ω00 at short times andtwice the Rabi frequency at long times (see also Figs. 9(d)

and 9(b)).

0 0.5 1 1.5 210

−15

10−10

10−5

100

hω/B

|F[

S(01)

π

]

|2

L = 21, η = 10, βDC = 1.9

(d)Squared modulus of Fourier transform of S(10)π

vs. rotationally scaled frequency for L = 21 sites andη = 10. Many new frequencies appear, in particularthe Rabi frequency and double the Rabi frequency,

denoted with arrows.

FIG. 9: Dependence of the asymptotic behavior of structure factors on the lattice height η.

49

example is Fig. 9(b), which clearly displays the 2Ω frequency behavior of the correlation structure factor S(01)π for

η = 10. This frequency, which we easily pick out in the site-averaged number’s Fourier transform, can also be seen in

the Fourier transform of S(01)π , see Fig. 9(d).

We find that the emergent time scale τ does not depend strongly on the size of the system L, even though thedistribution of molecules on the lattice is, in general, quite different for different numbers of sites, as can be seen bycomparing Figs. 6(a) and 6(e). Examining Fig. 6(c) and Table II, the L = 10 case has a smaller τ than either of theodd L cases. We think this has to do with the filling being exactly 1/2 and not, strictly speaking, with the numberof lattice sites, as the L = 9 and L = 21 cases have fillings less than 1/2. We see this clearly by comparing Fig. VD 2with Figs. 6(a), 6(c), and 6(e). Fig. VD2 displays 〈n00〉/N , a quantity which is independent of filling but dependent,in general, on the number of lattice sites. There is a weak dependence on the number of lattice sites. On the otherhand, Figs. 6(a), 6(c), and 6(e) display 〈n00〉/L, a quantity which is independent of the number of lattice sites butdependent, in general, on the filling. There is a marked difference between L = 10, which has filling of 5/10 = 1/2and the others, which have fillings< 1/2, but there is not a significant difference between L = 9 and L = 21, whichhave fillings of 4/9 and 10/21, respectively.

The dependence of τ on the filling is also evidenced by the correlation structure factor S(01)π in Fig. 7(f), which

shows that there is a stronger correlation between the J = 0 and J = 1 states for exactly half filling than for fillingsless than half, regardless of the system size. Half filling is known to be important in the extended Bose Hubbardmodel, where it marks the introduction of the charge density wave phase. We thus interpret this greater correlationstructure factor as the appearance of a dynamic charge density wave phase between rotational states at half filling.

This is in contrast to the usual behavior, where the structure factors S(00)π and S

(11)π are nonzero whenever there

is nonzero occupation of the particular rotational state and the structure factor S(01)π is much smaller–essentially

zero, see Figs. 7(a) and 7(e). These results for the structure factors means that the J = 0 and J = 1 states tendto lie on top of one another, and not to “checkerboard” with a different rotational state occupying alternating sites.This is due to the fact that the Rabi flopping time scale is much shorter than the dipole-dipole time scale, meaningthat the population cycles before there is sufficient time for the molecules to rearrange to a configuration which isenergetically favorable with respect to the dipole-dipole term. However, because the population in each rotational levelasymptotically reaches some nonzero value, we do see a small amount of rearrangement after many Rabi periods for

any filling, corresponding to a nonzero S(01)π . Note that this rearrangement does not affect the site-averaged numbers,

but rather the distribution of rotational states among the lattice sites. This asymptotic distribution emerges on timescales longer than we have considered, and is more prone to finite size effects than the site-averaged quantities, so wedo not make a conjecture about it here.

We find that the Q-measure saturates as

Q = Qmax − ∆Qe−tr/τQ , (302)

with a different time scale τQ, see Fig. V D2 and Table II. We also find that the saturation time scale of the Q-measureis not, in general, a monotonic function of the lattice height η, as shown in Fig. VD 2.

This time scale is different from the time scale τ at which the populations approach an asymptotic equilibrium,though both time scales respond similarly to changes in the Hamiltonian parameter, see Table II. For example, if τQgets larger as a parameter is changed then τ also gets larger, as illustrated in Figs. VD2 and 11(b). The time scaleτQ displays a stronger dependence on the number of lattice sites L than τ , as can be seen in Figs. VD 2 and VD 2.This is because τ describes a quantity that has been averaged over sites, while τQ does not.

VI. SEMICLASSICAL THEORY AND QUANTUM HYDRODYNAMICS

We now turn to a completely different class of emergent properties. So far we have focused on the fully quantumregime in which, for example, quantum fluctuations lead to a quantum phase transition. We studied various orderparameters and two-point correlations. We connected these two-point correlations to measurable response functionslike the dynamics susceptibility χ(k, ω). In this section, we will turn to the strong perturbation regime beyond linearresponse theory. We will look for new emergent phenomena which have energies far from the ground state.

To present this paradigm, we consider dilute weakly interacting scalar bosons. We avoid the lattice and insteadfocus on continuum properties directly, without first discretizing and then taking the long wavelength limit.

The second quantized Hamiltonian for weakly interacting bosons is

H =

∫

dV

[

ψ†(

− ~2

2M∇2 + V trap

)

ψ +1

2gψ†ψ†ψψ

]

, (303)

50

0 20 40 60 80 100 120 140 1600

0.2

0.4

0.6

0.8

1

1.2

Bt/h〈n

JM〉/N

L = 9L = 21L = 10

(a)Dependence of the population damping time scale τ on thenumber of lattice sites. When we remove the dependence on thefilling by dividing through by the total number, we see that there

is little difference in the time scales with which systems of differentsize approach dynamic equilibrium. Contrast Figs. 6(a), 6(c), and

6(e), which display a profound dependence on filling when thedependence on lattice sites has been removed.

0 50 100 1500.6

0.62

0.64

0.66

0.68

0.7

0.72

Bt/h

Q

η = 1, βDC = 1.9

L = 9L = 10L = 21

(b)Dependence of spatial entanglement on number of lattice sites.We see that systems of different size have different spatial

entanglement in their static ground state. The time scale of theQ-measure saturation, τQ, is shorter for L = 10 than it is for the

odd L cases. This follows the general trend of τ and τQ respondingcorrespondingly to changes in the Hamiltonian parameters, and sowe associate this shorter time scale partially with the filling, not

entirely with the system size.

FIG. 10: Dependence of emergent time scales on number of lattice sites.

where ψ(r) and ψ†(r) are annihilation and creation operators which obey the usual commutation relations:

[ψ(r), ψ†(r′)] = δ(r − r′), [ψ(r), ψ(r′)] = [ψ†(r), ψ†(r′)] = 0. (304)

In Eq. (303), V trap(r) is an external trapping potential and M is the mass of the bosons. The interaction term resultsfrom a short-range interaction g δ(r − r′), where g ≡ 4π~2as/M is a coupling constant with the dimensions of energy× volume and as is the s-wave scattering length. At low temperature and for a dilute gas only s-wave scattering isnon-negligible for bosons.

We consider the operator ψ(r, t) = exp(iHt/~) ψ(r) exp(−iHt/~) in the Heisenberg picture. Then it obeys theHeisenberg equation of motion

i~∂

∂tψ(r, t) = [ψ(r, t), H ] . (305)

51

0 20 40 60 80 100 120 140 1600.6

0.62

0.64

0.66

0.68

0.7

0.72

Bt/hQ

L = 10, βDC = 1.9

η = 1η = 2η = 3η = 4η = 10

(a)Dependence of spatial entanglement on lattice height. Note thatthe spatial entanglement and its associated time scale are not

monotonic functions of the lattice height. Note also that theentanglement of the static ground state appears to be largely

insensitive to the lattice height.

0 20 40 60 80 1000

0.1

0.2

0.3

0.4

0.5

0.6

Bt/h

〈n00〉

L = 10, βDC = 1.9

η = 1η = 2η = 3η = 4η = 10

(b)Dependence of the site-averaged number on the latticeheight. Note that the emergent time scale τ is not a monotonicfunction of the lattice height. Note also that τ responds in the

same way that τQ does to changes in the lattice height.

FIG. 11: Dependence of emergent time scales on lattice height.

Substituting Eq. (303) into Eq. (305), we find

i~∂ψ(r, t)

∂t=

(

− ~2

2M∇2 + V trap

)

ψ(r, t) + g ψ†(r, t) ψ(r, t) ψ(r, t). (306)

Next we make a crucial step related to the concept of Bose-Einstein condensation. In a non-interacting collectionof bosons we know from quantum statistical mechanics that all bosons will condense into the same mode. For weakinteractions, most bosons are still in a single mode: in the thermodynamic limit, the number of modes (in whateverbasis) approaches infinity. Therefore, on average the occupation of any given mode approaches zero; for bosons, solong as interactions are not too strong, one mode retains finite occupation. The criterion for weak interactions iscalled the dilute gas requirement:

√

na3s ≪ 1, where n is the average density. Provided this criterion is met, it can

be shown (much more formally than I am doing now) that the operator ψ can be replaced with a scalar:

ψ(r, t) = Ψ(r, t)1 + φ(r, t) (307)

where Ψ(r, t) is a classical field and φ(r, t) describes the remaining non-macroscopic modes. We sometimes call φ(r, t)quantum fluctuations ; in a statistical sense, this refers to the non-condensed particles. From this approximation weobtain the Gross-Pitaevskii equation or nonlinear Schrodinger equation (NLS) [54, 55]

i~∂Ψ(r, t)

∂t=

[

− ~2

2M∇2 + V trap + g |Ψ(r, t)|2

]

Ψ(r, t) . (308)

52

The scalar field Ψ(r, t) is the order parameter characterizing the thermal phase transition to a Bose-Einstein condensate(BEC). As this is a standard topic in statistical mechanics, we do not repeat it here. For the well-known atomic BECs,the NLS has been observed to provide an excellent description of most observable phenomena. We will focus on vortices

and solitons, two examples of emergent nonlinear phenomena. This is a sufficiently well-developed topic to have booksdevoted to it [7]. Equation (308) is sometimes called classical and sometimes semi-classical – we prefer the latter term.

If we require a separation of variables in time and space for the order parameter or wavefunction Ψ(r, t), we obtaina time-independent NLS:

[

− ~2

2M∇2 + V trap + g |Ψ(r)|2

]

Ψ(r) = µΨ(r) , (309)

where µ is a single particle energy, or chemical potential. We emphasize that for nonlinear equations the normalizationis non-trivial. Here

∫

V

|Ψ(r, t)|2 = N , (310)

where N is the number of particles, and we have taken N ≃ N0, the average number of particles in the condensate(since we have neglected the non-condensed modes).

A. Uniform Condensate

The stationary NLS, Eq. (309) has a natural length scale from the interaction term, known as the healing length,which is determined by dimensional analysis (engineering dimension):

ξ =~√

2Mng=

1√8πnas

, (311)

where n = n is the average density. In a free condensate with V trap = 0, this length characterizes the distance overwhich the condensate wave “heals” back to its bulk value when perturbed locally. A vortex is one such example of aperturbation.

Let us consider a condensate confined to a three-dimensional box of volume V . Then the condensate wave functionis

Ψ =

√

N

V. (312)

The energy can be calculated either directly from Eq. (303) together with the approximation of Eq. (307), or by directinspection of the stationary NLS of Eq. (309). The Hamiltonian functional in the canonical ensemble is

H =

∫

dV

[

Ψ∗(

− ~2

2M+ V trap

)

Ψ +1

2g|Ψ|4

]

, (313)

subject to the normalization constraint of Eq. (310). The grand canonical ensemble includes a Lagrange multiplierµ. Then in the 3D box, it is apparent that the ground-state energy Eg.s. comes only from the repulsive interparticleenergy of the condensate Eint ≈ 1

2 gN2/V . The chemical potential is

µ =

(

∂E0

∂N

)

V

= gn =4πas~

2n

M. (314)

The corresponding pressure follows from the thermodynamic relation

p = −(

∂E0

∂V

)

N

=1

2gn2 =

Eint

V. (315)

Finally, the bulk speed of sound cs follows from the compressibility:

c2s =1

M

(

∂p

∂n

)

=gn

M=

µ

M=

4πas~2n

M2, (316)

cs =~√

2Mξ. (317)

From Eqs. (311) and (316) it is apparent that a BEC is only uniform for repulsive interactions as > 0, since otherwisethe healing length and the speed of sound become imaginary. This corresponds to the well-known collapse problemfor the NLS, a wonderful study in applied mathematics [56].

53

B. Quantum Hydrodynamic Equations

First we make the Madelung transformation [57]

Ψ(r, t) = |Ψ(r, t)| eiS(r,t) . (318)

We interpret

n(r, t) = |Ψ(r, t)|2 , (319)

as the local condensate density. The corresponding current density is

j = (~/2Mi)[Ψ∗∇Ψ − (∇Ψ∗)Ψ] , (320)

or, in hydrodynamic form,

j(r, t) = n(r, t)v(r, t) , (321)

with an irrotational flow velocity

v(r, t) = Φ(r, t) (322)

expressed in terms of a velocity potential

Φ(r, t) =~S(r, t)

M. (323)

Next, to obtain fluid-like equations, we substitute Eq. (318) into the NLS, and break the equation into real andimaginary parts. From the imaginary part we get the continuity equation for compressible flow

∂n

∂t+ ∇ · (nv) = 0. (324)

The real part yields the analog of the Bernoulli equation for this condensate fluid

1

2Mv2 + V trap +

1√n

(

− ~2

2M∇2

)√n+ gn+M

∂Φ

∂t= 0. (325)

We can interpret this equation as follows [6]. First note that the assumption of a zero-temperature condensate impliesvanishing entropy, because there is only one microscopic configuration. Second, the Bernoulli equation for irrotationalcompressible isentropic flow can be rewritten as [58]

1

2Mv2 + U +

E + p

n+M

∂Φ

∂t= 0, (326)

where U is the external potential energy, E is the energy density and E + p is the enthalpy density. Comparison withEqs. (313) and (315) shows that Eq. (325) indeed has the appropriate relations for the enthalpy per particle,

(E + p)/n = (√n)−1

(

− ~2

2M∇2

)√n+ gn . (327)

As a result, the hydrodynamic form of the NLS in Eqs. (324) and (325), reproduces the standard hydrodynamicbehavior found for classical irrotational compressible isentropic flow. In particular, the dynamics of vortex lines atzero temperature follows from the Kelvin circulation theorem [58, 59]. This theorem states that each element of thevortex core moves with the local translational velocity induced by all the sources in the fluid, including self-inducedmotion for a curved vortex, other vortices, and net applied flow. The only clearly quantum-mechanical feature in

Eq. (325) is the quantum pressure (√n)−1

(

− ~2

2M∇2)√

n ; as seen from Eq. (311), this contribution determines the

healing length ξ that will fix the size and structure of the vortex core. In fact, Eq. (325) is equivalent to the Euler

equations for a classical inviscid (frictionless) fluid [60] with an added quantum pressure term, where the repulsiveinteractions provide the classical pressure and the quantum pressure only plays a role where the density is changing, asin the region of a vortex. Thus, instead of perfectly one-dimensional vortex lines, one finds some quantum uncertaintyin the region of the vortex core which specifies a core structure; in the study of classical systems, we would provide

54

some alternate model of a vortex core which would allow us to use the Euler equations far from the core and anothermodel in the core region, without having to solve the numerically difficult full-blown Navier-Stokes equations forviscous fluids.

In classical hydrodynamics it is reasonable to take the flow as incompressible when the velocity of the fluid is smallcompared to the speed of sound. Classical compressible flow becomes irreversible when the flow becomes supersonicbecause of the emission of sound waves; sound waves are included in the fluid equations themselves, and so do notrequire new deriviations and new mathematical descriptions. In a dilute Bose gas, Eqs. (324) and (325) neglect theuncondensed part entirely, according to our approximation made in Eq. (307). Instead, the system is unstable withrespect to the emission of quasiparticles once the velocity exceeds the Landau critical velocity, here given by cs. Thenormal component then plays an essential role and must be included in addition to the condensate. In this sense, adilute Bose gas is intrinsically more complicated than a classical compressible fluid; however, the vortex core model

is typically simpler. To get all of this right we have to at least go to first order in φ from Eq. (307).

C. Vortex Dynamics in Two Dimensions

Gross and Pitaevskii originally invented the NLS in order to study vortices, which are a key aspect of superfluidity.We follow the presentation from Fetter and Svidzinsky’s review [6] nearly verbatim in this section. Vortex solutionstake the form Ψ(r) =

√nχ(r), where n is the density far from the origin. Consider axisymmetric solutions of form

χ(r) = eiφf

(

r⊥ξ

)

, (328)

where (r⊥, φ) are two-dimensional cylindrical polar coordinates, and f → 1 for r⊥ ≫ ξ. Equations (322) and (323)give the local circulating flow velocity

v =~

Mr⊥φ, (329)

which represents circular streamlines with an amplitude that becomes large as r⊥ → 0.Equation (328) describes an infinite straight vortex line with quantized circulation

κ =

∮

dl · v =h

M. (330)

Stokes’s theorem then yields h/M =∫

dS · ∇ × v, with the corresponding localized vorticity

ω ≡ ∇× v =h

Mδ(2)(r⊥) z. (331)

Hence the velocity field around a vortex in a dilute Bose condensate is irrotational except for a singularity at theorigin.

The kinetic energy per unit length is given by

∫

d2r⊥ Ψ∗(

− ~2

2M∇2

)

Ψ =~2

2M

∫

d2r⊥ |∇Ψ|2 =~2n

2M

∫

d2r⊥

[

(

df

dr⊥

)2

+f2

r2⊥

]

. (332)

The centrifugal barrier in the second term forces the amplitude to vanish linearly within a core of radius ≈ ξ. Thiscore structure ensures that the particle current density j = nv vanishes and the total kinetic-energy density remainsfinite as r⊥ → 0. The presence of the vortex produces an additional energy Ev per unit length, both from the kineticenergy of circulating flow and from the local compression of the fluid. Numerical analysis with the NLS (numericalmethods for solution of the NLS are discussed further in Sec. VII) yields

Ev ≈ (π~2n/M) ln (1.46R/ξ ) , (333)

where R is an outer cutoff; apart from the additive numerical constant, this value is simply the integral of 12Mv2n.

To illustrate that the NLS describes the correct classical vortex dynamics, consider a state of the form

Ψ(r, t) =√n eiq·r χ(r − r0) e

−iµt/~, (334)

55

where χ is the previous stationary solution (328) of the NLS for a quantized vortex, now shifted to the instantaneousposition r0(t), and µ is now a modified chemical potential. The total flow velocity is the sum of a uniform velocityv0 = ~q/M and the circulating flow around the vortex. Substitute this wave function into the NLS Eq. (308). Sinceχ itself obeys the stationary NLS Eq. (309) with chemical potential µ = gn, a straightforward analysis shows thatµ = 1

2Mv20 + gn, where the first term arises from the center of mass motion of the condensate. The remaining terms

yield

i~∂χ(r− r0)

∂t≡ −i~dr0

dt· ∇χ(r − r0) = −i~v0 · ∇χ(r − r0). (335)

This equation shows that dr0(t)/dt = v0, so that the vortex wave function moves rigidly with the applied flow velocityv0, correctly reproducing classical irrotational hydrodynamics.

A similar method applies to the self-induced motion of two well-separated vortices at r1 and r2 with |r1 − r2| ≫ ξ.In this case,

Ψ(r, t) =√nχ(r − r1)χ(r − r2) e

−iµt/~ (336)

represents an approximate solution with µ = ng because there is no net flow velocity at infinity. The densityn |f(r− r1)|2|f(r− r2)|2 is essentially constant except near the two vortex cores, and the phase is the sum S(r − r1)+S(r − r2) of the two azimuthal angles for the variable r measured from the local vortex cores. Substitution into thetime-dependent GP equation readily shows that each vortex moves with the velocity induced by the other, for example

dr1

dt≈ ~

M∇S(r − r2)

∣

∣

r=r1. (337)

This method also describes the two-dimensional motion of many well-separated line vortices [61].

VII. NUMERICAL METHOD I: PSEUDO-SPECTRAL ADAPTIVE TIME-STEP RUNGE KUTTA

Note: This section is written to be entirely self-contained, in case you should wish to study solitons in BECs as a

stand-alone topic. Thus a few equations will be repeated from Sec. VI. Since these are in any case key equations, it

is good to see them twice, even if you are reading these notes from start to finish.

Let us proceed to study the simplest case for quantum hydrodynamics, that of one spatial dimension. Clearly wecan adapt the uniform solution of the NLS from Sec. VI A to one dimension in order to obtain the ground state. Whatabout the promised emergent phenomena with an energy far from the ground state? In the case of one dimensiononly, we can do everything in closed analytic form for stationary states. However, for the ensuing dynamics we requirenumerical methods. Thus this section bridges the gap between analytical and numerical methods.

A. Solitons and stationary states

We consider two cases of the BEC: repulsive and attractive atomic pair interactions. For our purposes they differby the sign of the nonlinear term in the NLS. The former is the one that has received the most experimental attention.The latter is generally unstable in 3D [62] but stable in 1D [63].

The NLS in 1D has a number of special properties which are described in the mathematical literature. It isintegrable [56, 63], may be solved exactly by the Inverse Scattering Transform [64, 65] and has a countably infinitenumber of conserved quantities [66, 67]. We have taken advantage of its special properties in finding its stationarysolutions. We first describe all stationary states of the 1D NLS subject to box and periodic boundary conditions. Wethen make clear their connection with solitons and provide physical interpretations of their form.

1. Mathematical form

We begin with the NLS that describes a BEC of N atoms of mass M , confined in an external potential V (~r):

[− ~2

2M∇2 + g |ψ(~r, t) |2 +V (~r) ]ψ(~r, t) = ı~∂t ψ(~r, t) (338)

56

0 0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

0

0.5

1.0

1.5

2.0

(c)

(a) (b)

0 0.2 0.4 0.6 0.8 1

-1

-0.5

0

0.5

1

(d)

0 0.2 0.4 0.6 0.8 1

-1.5

-1

-0.5

0

0.5

1

1.5

0 0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

0

0.25

0.5

0.75

1.0

antisymmetricsymmetric

phaseamplitude

amplitude

x

phase

x

x x

FIG. 12: Shown are the five stationary solution-types in one dimension under periodic boundary conditions. Repulsive case: (a)amplitude of real solution and (b) amplitude and phase of complex solution. Attractive case: (c) amplitude of real symmetricand antisymmetric solutions and (d) amplitude and phase of complex solution. All plots are normalized to the unit interval.

where |ψ(~r, t)|2 is the single particle density such that ρ(~r, t) = N |ψ(~r, t)|2, the coupling constant g ≡ 4π~2aN/M ,and a is the scattering length for binary collisions between atoms. The case of repulsive interactions corresponds toa > 0; that of attractive interactions to a < 0.

The characteristic length scale of variations in the condensate wavefunction is the healing length ξ:

ξ ≡ (8πρ |a |)−1/2 (339)

where ρ is the mean particle density. The BEC is in the quasi-1D regime when the lengths of the tranverse y andz dimensions of the trap, Ly and Lz, satisfy the following criteria: Ly, Lz ∼ ξeff and Ly, Lz << Lx. The formerensures that the condensate remains in the ground state in the two transverse dimensions while the latter ensuresthat longitudinal excitations are much lower in energy than possible transverse excitations. Under these conditionsone may make an adiabatic separation of longitudinal and transverse variables [8, 68]. Equation (338) then reducesto:

[−(ξeff )2∂2x± |f(x) |2 +V (x) ] f(x) = ı∂t f(x) (340)

where all terms have been made dimensionless: f(x) is the longitudinal portion of the wavefunction; t ≡ (~/2Mξ2)t

we will call the natural time; the ± refer to the repulsive and attractive cases, respectively; V (x) ≡ (2Mξ2/~2)V (x);and ξeff is an effective healing length, discussed in section VII C 1. If we furthermore assume a stationary state:

[−(ξeff )2∂2x± |f(x) |2 +V (x) ] f(x) = µ f(x) (341)

where µ ≡ (2Mξ2/~2)µ is a dimensionless chemical potential which is now an eigenvalue.All stationary solutions to equation (341) may be written in terms of Jacobian elliptic functions [69]. The properties

of such functions are reviewed elsewhere [8, 69, 70]. There are five normalizable symmetry-breaking solution-typesto the stationary NLS with a constant potential on a finite interval. They are pictured in figure 12. The stationarysolutions appropriate to longitudinal box boundary conditions are:

57

f(x) = A sn(2jK(m)x

Lx+ δ | m) (342)

f(x) = A cn(2jK(m)x

Lx+ δ | m) (343)

for the repulsive and attractive cases, respectively. Equation (342) is shown in figure 12(a) and equation (343) isshown as the anti-symmetric solution in figure 12(c). A is the amplitude, j − 1 is the number of nodes, K(m) is thecomplete Jacobian elliptic integral which is the quarter period of these functions, δ is the offset, here equal to zero,and 0 ≤ m ≤ 1 is the Jacobian elliptic parameter. The general notation sn(u | m) is standard for Jacobian ellipticfunctions [69, 70].

Under periodic boundary conditions equations (342) and (343) are still solutions, but j is required to be even andthe number of nodes is j, i.e. j = 2, 4, 6, ... Translational symmetry is restored, so the offset δ is no longer required tobe zero. This leads to a Kosterlitz-Thouless-type entropy in 1D [8].

In addition to the above two solution-types there are three nodeless, symmetry-breaking solution-types underperiodic boundary conditions. The first, a solution to the attractive case, is real:

f(x) = Adn(2jK(m)x

Lx+ δ | m) (344)

An example is shown as the symmetric solution in figure 12(c). The other two types are intrinsically complex.f(x) ≡ r(x) exp(ıφ(x)) and for the repulsive and attractive cases, respectively:

r(x)2 = A2(1 − γ dn2(2jK(m)x

Lx+ δ | m)) (345)

r(x)2 = A2(dn2(2jK(m)x

Lx| m) − γ(1 −m)) (346)

where in both cases the phase must be found by numerical integration from the equation:

φ′(x) =α

r(x)2(347)

The phase and amplitude are shown in figure 12(b) and 12(d). In the repulsive case A2γ is the depth of the densityminima below the constant background. When γ = 1 equation (342) is recovered. In the attractive case γ interpolatesbetween the real anti-symmetric and symmetric solutions in equations (343) and (344). In both cases 0 ≤ γ ≤ 1. αis a constant of integration. j is the number of density minima or maxima, respectively.

All five solution-types are found by solving equation (341) subject to normalization and boundary conditions, andare described in detail in references [8] and [9].

2. Physical interpretation

The nonlinear mean field term in equation (341) causes a spreading out or a clumping together of the wavefunctionbetween nodes for the repulsive and attractive cases, respectively. Thus the stationary solutions to the NLS with boxboundary conditions are in one-to-one correspondence with the sinusoidal solutions to the particle-in-a-box problemin linear quantum mechanics. The number of nodes is j − 1 and the complete set of stationary states is described byequations (342) and (343), where j ∈ 1, 2, 3, ....

Likewise, if j is restricted such that j ∈ 2, 4, 6, ..., equations (342) and (343) are in one-to-one correspondencewith the sinusoidal solutions to the particle-on-a-ring problem in linear quantum mechanics. As the ring has rotationalsymmetry these solutions are symmetry-breaking. We note that, as in the linear Schrodinger equation, there are alsocomplex, constant-amplitude, plane-wave solutions on the ring.

The Jacobian elliptic parameter m governs the strength of the nonlinearity. As m → 0+, sn → sin and cn → cos,respectively. This is the linear, sinusoidal limit. As m → 1−, sn → tanh and cn → sech. These are the dark and

58

bright soliton, stationary solutions to the NLS on the infinite line, respectively [64, 65]. This shows the connectionbetween these stationary solutions and solitons. In the box the jth stationary state is a j − 1 or j soliton-train inthe repulsive and attractive cases, while on the ring the 2jth stationary state is a 2j soliton-train. As we have shownelsewhere [11], soliton-trains interact in a manner similiar to single solitons. And as with single solitons their formand phase relationships are quite robust.

The other three solution-types have no analogue in the particle-on-a-ring problem in linear quantum mechanics.They exist only on the ring and are nodeless and symmetry-breaking. The complex solutions have a monotonicallyincreasing phase. Each has a complex-conjugate, degenerate partner. The complex solutions for the attractive case,in particular, are an entirely new stationary solution-type for the NLS.

The complex solutions for the repulsive case are interpreted as density-notch solitons moving with speed c on thering with an opposing momentum boost of the condensate of speed −c, which results in a stationary state in the labframe. Density-notch solitons have a speed between zero and the Boguliubov sound speed, ranging from maximal tozero depth, respectively [71]. Those not of maximal depth are called grey solitons, while those which are of maximaldepth and therefore form a node are called dark. Figure 12(b) shows the bounded, quantized version of a grey2-soliton-train.

At typical experimental trap sizes the single density-notch stationary state on the ring is the lowest energy,symmetry-breaking excitation above the real, constant-amplitude ground state. When any of these stationary excitedstates is perturbed it gives rise to soliton-type motions [71]. Recent reports suggest that such motions can be inducedin repulsive BECs by optical phase engineering of the condensate phase [72, 73].

All attractive symmetry-breaking, longitudinally periodic, stationary solution-types, i.e. the anti-symmetric andsymmetric ones shown in figure 12(b) and the complex one shown in figure 12(d), are described by the Cj pointsymmetry group, where j is the number of peaks. There are j nearly degenerate solutions. For even j there is a realsymmetric-anti-symmetric pair and (j − 2)/2 degenerate complex pairs. For odd j there is a real symmetric solutionand (j − 1)/2 degenerate complex pairs. Thus by group theory these three solution-types form the complete set ofstationary states made of evenly spaced peaks.

As the attractive BEC has been made in 3D but not in quasi-1D, these stationary solutions are open to experimentalinvestigation.

3. Phase engineering of dynamics

There are two ways to impart velocity to a density-notch soliton. One may boost the constant background viaa phase ramp and the notch will drift with the resulting super-current. One may also apply a phase jump and theresulting soliton speed with respect to the background will be c = cmax sin(δ/2), where cmax is the Boguliubov soundspeed and δ is related to the phase difference across the moving density notch [71]. Both types of phase profile areapparent in figure 12(b). The phase ramp may be seen in the constant background slope in the phase. The phasejump is shown over each of the two notches. The velocities cancel and a stationary state results.

To impart a collective velocity to trains of density-notches one applies an equal phase difference across each of themember notches or a phase ramp across the whole train. One may also treat the members of the train individuallyby applying varying jumps and thereby study solitons interactions. A bright soliton may be set in motion by a phaseramp but not a phase jump. For a phase ramp exp(ıkx) the resulting velocity c ∝ k. There is no upper bound onthe velocity of bright solitons. As with density-notches, a train of bright solitons may be manipulated collectively byapplying a phase ramp across the whole set or individually by applying different phase ramps across the members ofthe train.

Soliton-trains are thus collective excitations which can be directly manipulated by simple phase profiles. One maymanipulate the members of the train individually or collectively. This makes clear the connection between the abovestationary states and solitons.

B. One dimension

1. Stability

We first demonstrate that the five solutions types described in section VII A 1 are indeed stationary. In figure 13 wepropagate them in time numerically. In the numerical work presented for one, two, and three dimensional solutionsof the NLS the condensate wave function was expanded as ψ(~r, t) = Σici(t)φi(~r). The φi(~r) were chosen as a co-ordinate-discretized, pseudo-spectral basis, and the ci(t) were propagated in time using a fourth order variable-step

59

> >>

(f)

(h)

(j)(i)

DensityPhase

x x

t

(c)

(e)

(a)

(g)

(b)

(d)

min max2π0

FIG. 13: The five stationary solution-types in one dimension under periodic boundary conditions are propagated in timenumerically for 100 natural time units. Shown are phase and density for (a)-(b) real solutions, repulsive case; (c)-(d) complexsolutions, repulsive case; (e)-(f) anti-symmetric real solutions, attractive case; (g)-(h) symmetric real solutions, attractive case;(i)-(j) complex solutions, attractive case.

Runge-Kutta algorithm, made highly efficient by implementation with spatial fast Fourier transform techniques. Weshall use the same numerical methods throughout our article.

The denumerably infinite number of conserved quantities in the 1D NLS makes solitons robust. Single solitons, inwhatever equations they arise, are in general stable. Here in both the repulsive and attractive cases soliton-trainsare stable [11]. In the attractive case the two nodeless solution-types are cyclically stable. That is, in response tostochastic perturbation they evolve in a complicated manner but come asymptotically close to their original statein an aperiodic cycle. In figure 14 we illustrate the stability properties of the stationary states by introducing 1%

60

FIG. 14: The five stationary solution-types in one dimension under periodic boundary conditions are propagated in timenumerically, as in figure 13, for 100 natural time units. Stochastic noise at the level of 1% is added to the initial state. (a)-(f)are stable; (g)-(j) are cyclically stable. Shown are phase and density for (a)-(b) real solutions, repulsive case; (c)-(d) complexsolutions, repulsive case; (e)-(f) anti-symmetric real solutions, attractive case; (g)-(h) symmetric real solutions, attractive case;(i)-(j) complex solutions, attractive case.

stochastic white noise into all five fundamental stationary solution types and observing their evolution.Stochastic perturbation of density-notch solitons can cause a velocity shift or small-scale emission of phonons [63].

The phase relationships between solitons, however, remain intact. The depth of a density-notch is a simple functionof its velocity. Therefore the soliton-train changes in depth but not in spacing or form. It drifts as a unit in a diffusivemanner. This is true for both the real and complex solution-types.

Stochastic perturbation of bright solitons can cause a velocity shift, small-scale emission of phonons, and/or a phase

61

shift [74]. Although a single soliton is stable, multiple solitons are cyclically stable. As the phase difference betweenadjacent solitons changes from zero to π their interaction changes from attractive to repulsive [11], and it is thischanging interaction that leads to cyclical stability. The anti-symmetric solution is stable. The repulsive interactionsbetween solitons lock them into place. In the other two solution-types the solitons attract and repel as their relativephases are perturbed by the phonons that make up the white noise. We have not shown their recurrence here.

2. Expectations in higher dimensionality

We expect that in quasi-1D the five solution-types described analytically in section VII A 1 and numerically insection VII B1 will have the same properties as in 1D. We also expect, and will verify in the following section, thatexcellent quasi-1D stationary states can be found as the direct product of eigenstates of the 1D NLS in each of thedimensions, even though the NLS is only approximately separable.

To reiterate our criteria for quasi-1D from section VII A 1: Ly, Lz ∼ ξeff ; Ly, Lz << Lx. In the repulsive case anon-overlapping 2-soliton solution becomes possible when the length is 2πξeff [8]. This provides a criterion for thebreakdown of the quasi-1D approximation: if Ly, Lz > 2πξeff a transverse excitation becomes likely.

Solutions to the attractive NLS become unstable when the negative mean field energy is not sufficiently opposedby the positive kinetic energy. This may be seen by performing a variational calculation. So one criterion we coulduse is the length for which the transverse solution changes from positive to negative energy. This is about 2 1

2ξeff . Inthe literature it has been found that the 3D attractive NLS is more likely to go unstable than the 2D NLS [75]. Sowe expect that the quasi-1D approximation will break down at a smaller transverse length in 3D than in 2D. In theattractive case we expect that out of the quasi-1D regime the wavefunction should collapse.

We note that in the repulsive case the criteria for separability and quasi-1D are not identical. In the limit as thetransverse lengths become very large the NLS becomes exactly separable for a constant transverse solution. However,the separable solutions can be unstable, as we shall show in section VII D.

In the repulsive case it has been shown in the mathematical literature that band soliton and planar soliton solu-tions of the NLS in 2D and 3D can decay into vortices via long-wavelength transverse modulations known as snakeinstabilites [63, 76]. They are called band or planar solitons both to describe their shape in 2D and 3D and todifferentiate them from 1D solitons of the type formally obtained by the Inverse Scattering Transform [64, 65]. Thusas we increase the length of the transverse dimensions out of the quasi-1D regime and into 2D and 3D we expect tofind that our soliton-train solutions break up into vortices. The BEC conserves vorticity. When the band or planarsolitons decay the circulation quantum numbers of the resulting vortices should add to zero. Since the energy of avortex is proportional to its quantum number squared we expect the decay to be into pairs of singly-quantized vorticesof opposite circulation.

C. Quasi-One-Dimension

To demonstrate the experimental viability of the 1D stationary states described above we add one or two transversedimensions of scale size ξeff and study the stability and phase engineering of solitons and stationary states in quasi-1D. The connection between our numerical studies and present BEC experiments is made clear. We also illustrate thephase rigidity of the quasi-1D BEC. We hope the following work will encourage experimentalists to create a quasi-1Dattractive BEC and to study density-notches and phase rigidity in the quasi-1D regime of the repulsive BEC.

1. Connection with Experiments

In table III we show the length and time scales and the number of atoms for the four atomic species of BEC. Thebelow conversion formula allows one to find the number of atoms given the healing length, trap size, and type of BEC.

N =ξeffLxLyLz

8πa(348)

where the healing length ξeff and the scattering length a must be in the same units and the trap dimensions Lx, Ly,and Lz are in units of ξeff . From section VII A 1 the time scale is:

t =2Mξ2eff

~t (349)

62

TABLE III: Physical time and length scales of our numerical studies. Thus 100 time units is 20-200 ms. Note that the numberof atoms scales with the trap volume, so that 104 atoms would be found in the quasi-1D regime of current experimental volumes.We fix the healing length at 3 µm for the sake of observability.

atom V (ξ3eff ) ξeff (µm) t (ms) N as (nm)

23Na 1x1x25 3.0 0.727 1,610 2.75

87Rb 1x1x25 3.0 2.75 769 5.77

1H 1x1x25 3.0 0.0316 76,500 0.0581

7Li 1x1x10 3.0 0.221 1,250 1.45

where M is the mass of the atomic species. t are the natural, unitless time units we use in the numerical studies.In the table we choose ξeff = 3.0µm to make solitons in the condensate easily observable. This is the order of the

healing length used in recent soliton experiments [72, 73]. The time scales are on the order of 0.1 to 1 ms per naturaltime unit. Since we present results on time scales from 25 to 100 natural time units all our numerics take place onthe experimental time scales of from 1 to 100 ms. Similiar results persist to 1000 natural time units. As BECs lastfrom 1 to 75 s our studies extend over the lifetime of current BECs [77, 78].

For computational reasons we have used somewhat smaller traps than current experiments, 10x1x1 or 10x 12x 1

2 inthe attractive case and 25x1x1 in the repulsive case. Thus the number of atoms is small. In the attractive casethe trap can be made longer and remain in the quasi-1D regime as long as the number of peaks in the solutions isincreased. In the repulsive case the trap size could be increased in each transverse dimension by a factor of 5 andarbitrarily in the longitudinal dimension, resulting in a factor of 100 or more increase in the volume. Therefore thenumbers we have used in the simulations scale into the experimentally accessible regime.

The assumption of separability in quasi-1D is made by projecting the total wavefunction onto the ground states ofthe transverse dimensions and integrating. This produces an effective healing length which depends on the transversewidth:

ξ2eff = (3 (E(mt)/K(mt) − 1)2

−2(1 +mt)E(mt)/K(mt) − (2 +mt))2 ξ2 (350)

ξ2eff = (3 (E(mt)/K(mt) − (1 −mt))

2

2(−1 + 2mt)E(mt)/K(mt) − (2 − 5mt + 3m2t )

)2 ξ2 (351)

for the repulsive and attractive cases, respectively. mt is the Jacobian elliptic parameter which determines the widthof the transverse wavefunction, which is a sn or cn function described by equations (342) or (343). E(mt) and K(mt)are complete Jacobian elliptic integrals.

In the limit as mt → 0+, ξ2eff → 49ξ

2 in both cases and the transverse wavefunction is sinusoidal. As mt → 1−,

ξ2eff → ξ2 and ξ2eff → 0 monotonically in the repulsive and attractive cases, respectively. For the purposes of the

present quasi-1D numerical investigations we shall use a transverse healing length of 1 or 12 . For these healing lengths

mt ∼ 0.1 and ξ2eff ∼ 49ξ

2. We have used these rescalings in determining the number of atoms in table III.Besides the ratio of the box lengths to the healing length, the numerical studies do not depend on the choice of

the experimental parameters. In all figures we have set ξeff = 1 and thus are working in length units scaled to theeffective healing length.

2. Stability

As in section VII B 1 we again add 1% stochastic white noise to the five solution-types, but in both the 2D and 3Dquasi-1D limits. The initial state is created from the product of box transverse wavefunctions of the form describedby equations (342) and (343) with a box or periodic longitudinal wavefunction of each of the five types shown infigure 12.

In figure 15 we show results of simulations in the 3D case. Trap parameters which enforce the quasi-1D dynamicsare used: a volume of 25x1x1 in the repulsive case and 10x 1

2x 12 in the attractive case. The stability properties are

identical with those found in 1D. Although we have not shown it here, the 2D case displays essentially the sameresults.

63

>x >x

Density

>

Phase

>

(i)

(b)

(d)

(f)

(h)

(j)

(a)

(c)

(e)

(g)

12.5

t=6.25

25

0 2π min max

y

37.5

25

t=12.5

50

18.75

FIG. 15: The five stationary solution-types are extended into quasi-one-dimension under periodic boundary conditions. Theyare propagated in time numerically as in figure 14, for 50 and 25 natural time units in the repulsive and attractive cases.Stochastic noise at the level of 1% is added to the initial state. (a)-(f) are stable; (g)-(j) are cyclically stable. Shown arefour time slices through the xy plane of phase and density of fully 3D simulations for (a)-(b) real solutions, repulsive case;(c)-(d) complex solutions, repulsive case; (e)-(f) anti-symmetric real solutions, attractive case; (g)-(h) symmetric real solutions,attractive case; (i)-(j) complex solutions, attractive case.

64

The repulsive solutions drift but are otherwise unchanged. The complex solution-type moves with a steady velocityto the left because the approximation of separability has failed to give the right momentum boost of the condensateto counteract the motion of the soliton-train. But clearly a larger momentum boost would return it to a stationarystate. Note that the phase relationships are quite stable: constant between notches in figure 15(a) and changinglinearly between notches in figure 15(c).

The attractive anti-symmetric solution is locked into place and is stable as it was in 1D. The two nodeless solution-types break up and recur. The symmetric, real type shown in figure 15(g)-(h) shows an example of this recurrence.In the third time slice the left-hand peak has spread out and changed its phase relationship to the right-hand peak.In the fourth time slice it has returned to its initial state.

As predicted, there are small fluctuations due to the assumption of separability in our choice of initial state. Inthe repulsive case we found that beyond six healing lengths the solutions became unstable in both 2D and 3D. In theattractive case we found that the requirements for the quasi-1D regime were more complicated. A transverse lengthof 1

2ξeff was required in 3D while ξeff was sufficient in 2D. In general the stability depended both on the numberof peaks and the solution-type. The anti-symmetric solution-type with a large number of peaks was the most stableconfiguration. This kept the density spread out and the contribution of the mean field term in the NLS small. Otherresearchers have found similiar results [79].

3. Manipulation Via Phase

In section VII A 3 we discussed simple phase profiles which impart a velocity to soliton-trains. Setting the wholetrain in motion around a thin torus creates a stable, structured super-current in both repulsive and attractive cases.Measuring the velocity of the train is a way to verify the superfluidity of the gaseous BEC. One method would becombination of the new toroidal trap, created to Bose condense Cs [80], with optical phase-imprinting [81]. Note thatthe superfluid properties of the attractive BEC are presently unknown.

One may study soliton interactions by treating members of the train separately. In figure 16 we show that planarsolitons in quasi-1D interact in a manner similiar to solitons in 1D. In the repulsive case we have used an 0.3π phasejump to send two density notches towards one another. As may be seen in the figure, they undergo elastic repulsiveinteraction twice over 50 natural time units.

In the attractive case we have used a real symmetric initial state with a phase ramp of 2π on each bright soliton.Their first and third interactions show a characteristic smearing of the density which indicates a phase differencebetween zero and π. In the second interaction they add coherently as they pass through each other, indicating aphase difference of zero. In their fourth interaction they repel. Therefore it is quite apparent that their relative phasehas drifted by approximately π, due to the 1% stochastic noise added to the initial state.

For computational reasons we chose phase profiles which caused interaction on a short time scale. But similarinteractions could be observed on longer time scales by the appropriate choice of phase jump or ramp. Thereforesolitons and stationary states in quasi-1D are connected in the same way as in 1D. Soliton interactions can be studiedexperimentally in the BEC.

By use of phase ramps one may also study how the BEC responds to shock. The repulsive case has the propertyof phase rigidity, related to symmetry-breaking of the condensate phase. Rigidity is common to many symmetry-breaking materials in condensed matter physics, from liquid crystal to chalk [82]. The BEC exhibits this property byshattering into well-defined phase domains, the size of which are on the order of the healing length. In quasi-1D theborders between domains are made of density-notch solitons.

To illustrate this property we use longitudinal box rather than periodic boundary conditions, with volume 25x1x1.A large phase ramp of φ(x) = 1.6πx is applied to a two-density-notch stationary state. In figure 17 we show thetime evolution of such a configuration. The first time slice shows the initial state. The second shows a shock wave,apparent in the density. The final time slice shows a highly excited state with well-defined phase domains separatedby density notches. Because the kinetic energy of this state is very high the nonlinear term in the NLS does notcontribute significantly. Thus a linear beat phenomenon is seen in the density. We have used short time scales forconvenience, but a smaller phase ramp would cause the same phenomenon to occur on a much longer, experimentallyobservable time scale.

D. Beyond quasi-one-dimension

In this section the transverse dimensions are extended beyond the quasi-1D regime in both two and three dimensions.Band and planar solitons become unstable, and shocking the condensate produces chaos.

65

Phase

>>

t

x> >x

(d)

>

Density

(a) (b)

(c)

0 2π

y

min max

FIG. 16: In quasi-1D soliton-train solutions can be manipulated by simple phase profiles. Here we show examples of suchmanipulation for (a)-(b) the repulsive and (c)-(d) the attractive cases. The solitons have been given equal and oppositevelocity. The evolution of their phase and density is shown by plotting the xy plane at the midpoint of the z co-ordinate intime slices running down the page. We have added 1% stochastic noise to the initial state. The results shown follow from afully 3D propagation with dimensions 25x1x1 in the repulsive case and 10x 1

2x 1

2in the attractive case.

1. Vortex creation

In figures 18 and 19 we show the decay of band and planar solitons into vortices. In both cases we begin with2-soliton-train stationary states made in the same way as described in section VII C. We use dimensions 25x12 in 2Dand 25x12x12 in 3D with periodic boundary conditions in the longitudinal direction. White noise causes the solitons

66

Phase

>

15

10

5

(b)

Density

>x

(a)

>

20

x

2π min0 max

t=0

y

FIG. 17: A rapidly translating condensate in a quasi-1D box is suddenly stopped. The condensate shatters into domains ofconstant phase due to the fragility of this phase-symmetry-breaking system.

Phase

>

48

(b)(a)

60

Density

>x

72

>

36

x

2π min0 max

t=24

y

FIG. 18: In 2D band solitons decay into pairs of vortices of the same charge. We have added 0.01% stochastic noise to aninitial two-soliton stationary solution in order to demonstrate this effect. Shown are five slices of the xy plane at the midpointof the z co-ordinate, equally spaced in time. The longitudinal direction has periodic boundary conditions. Note that the phaseleads the instability.

67

Phase

>

30

(b)(a)

24

Density

>x

18

>

36

x

2π min0 max

t=12

y

FIG. 19: In 3D planar solitons decay via a snake instability. We have used the same noise level as in figure 18. We havehighlighted an earlier set of times, as the decay occurs earlier than in 2D. Shown is the xy plane at the midpoint of the zco-ordinate.

to decay into a pair of vortices. The vortices are singly quantized, as may be seen in the figures by traversing a nodein the phase. The colors pass once around the phase color circle, which is defined to be 2π in extent. In the plots wechose times that would highlight the decay. However, the vortices are stable, spatially stationary, and do not interactover the full length of the study.

White noise in the BEC is principally due to uncondensed atoms and the number of such atoms is a function oftemperature. Thus noise is nominally an experimental parameter. The time from creation of the initial stationarysolitons to their decay into vortices is a function of the noise. With the correct choice of temperature and volume,stationary band or planar solitons, made via phase engineering [12, 73], can be used to create vortices in a controlledmanner.

2. Vortex interaction

Band and planar solitons respond to simple phase profiles in the same way as solitons in quasi-1D. The followingmethod permits direct study of vortex interactions.

As explained above, the time from creation of the initial state to decay into vortices is a function of the temperatureof the BEC. As in quasi-1D, they move toward each other with a velocity which depends on the phase jump. Thustwo solitons can be made to decay into vortices shortly before collision. An example of this process in 2D is shown infigure 20. We use a phase jump of π/15, a noise level of 0.1%, and a container identical to the one in figure 18. In thefirst time slice a snake instability and a vortex pair have developed. By t = 36 the vortices are strongly interacting.They have become a pair of vortex dipoles. By t = 48 they have completed their interaction and are moving awayfrom each other around the torus. The time slices in the figure are evenly spaced. Thus it can be seen that, unlike inquasi-1D, the collision is not elastic. The relative vortex velocity in the first three time slices, before the collision, isnot the same as in the last two. In fact the two vortex dipoles no longer have the same speed.

Although we have not pictured it here, the evolution of the above collision is very intriguing. Despite the complicateddensity profile, it is apparent from the phase that the remnants of the two band solitons oscillate between singlevortices, vortex dipoles, and band solitons. The relationship between the spectra of these various collective excitationsis an unsolved problem.

68

Phase

>

42

(b)(a)

36

Density

>x

30

>

48

x

2π min0 max

t=24

y

FIG. 20: Collision of two initial band solitons which have evolved into two oppositely charged vortex pairs. We have added0.1% stochastic noise. As in figure 16 the relative velocity is induced by phase discontinuities across the band solitons, but theinteraction is that of two vortex dipoles. This is an example of how to use the noise-induced instability of band solitons in 2and 3D to study vortex interactions.

3. Shock and phase rigidity

Explorations of 2D and 3D out of the quasi-1D regime have thus far have been with periodic boundary conditionsin the longitudinal direction. If instead box boundary conditions are used, together with a phase ramp of the kinddesribed in the previous section, the BEC becomes chaotic. As in quasi-1D, it exhibits the property of phase rigidity.In response to shock it breaks up into well-defined phase domains.

To illustrate this effect a volume of 25x12 with box boundary conditions in both dimensions was used. A largephase ramp of φ(x, y) = 1.6πx+ 0.2πy pushed a two-soliton, approximate stationary state into the walls at a skewedangle. The time evolution of the resulting state is shown in figure 21. The first time slice shows the initial state. Thesecond time slice shows a shock wave. By the final two time slices the condensate appears to have fractured into achaotic assortment of phase domains. The phase appears to be completely decoherent at scales larger than the healinglength.

VIII. NUMERICAL METHOD II: EXACT DIAGONALIZATION

In exact diagonalization we solve the full Hamiltonian, typically with the aid of a computer. Quantum mechanicsin this approach comes down to a very large linear algebra problem. There are standard codes, such as those found inthe ALPS library, to perform such computations [16, 17]. In the Appendix we provide a problem set, together withsolutions, for teaching the techniques of exact diagonalization starting from a very basic level.

A. Bose-Hubbard Hamiltonian in a Rotating Frame

As an example of the utility of such methods, we discuss solution of the Bose Hubbard Hamiltonian in a rotatingframe. This maps onto the quantum Hall problem, since rotation is equivalent to a magnetic field. The general

69

Phase

>

3.75

(a) (b)

2.5

Density

>x

1.25

>

5.0

x

2π min0 max

y

t=0

FIG. 21: A rapidly translating quasi-2D repulsive condensate is stopped. Following propagation of a density-dependent shockfront, a fully chaotic phase and spatial distribution is obtained.

Hamiltonian is [83]:

H = −t∑

〈~ri,~rj〉

a†(~ri)a(~rj)ei∫ ri

rjd~r· ~A(r)

+ h.c.

+∑

~ri

Vex(~ri) − V cf(~ri) − µ

n(~ri)

+U

2

∑

i

n(~ri)(n(~ri) − 1) +W∑

〈~ri,~rj〉n(~ri)n(~rj),

~A(~r) =m

~

~Ω × ~r, V cf(~ri) = −1

2mΩ2r2i , (352)

where µ is the chemical potential, ~Ω = Ωz is the rotation angular velocity, ~r is the position vector with respect to

the axis of rotation, the vector potential ~A is due to the Coriolis force, V cf is the centrifugal potential, and V trap is

the trapping potential. U is scaled to 1. For the charged bosonic system in the magnetic field ~B, ~A = (e∗/2~c) ~B × ~rinstead and the V cf term is absent.

We consider a slightly easier case in which the exponentials involving the vector potential are expanded to lowestorder. We follow our previous work in this discussion [18]; a more complete discussion can be found in Ref. [84].

Consider bosons in 2D, interacting via a two-body contact potential of strength g in a 2D lattice potentialV lat(x, y) = V lat(x + jd, y + kd) which rotates about the z axis, where j, k are integers and d is the lattice con-stant. In the rotating frame the Hamiltonian is

H =

∫

d2r Φ†[

− ~2

2M∇2 +

g

2Φ†Φ + V lat − ΩLz

]

Φ , (353)

Lz ≡− i~(x∂y − y∂x) (354)

with rotation frequency Ω and atomic mass M , where Φ†(x, y) and Φ(x, y) are field operators obeying the usualbosonic commutation relations. Using a Wannier basis, the operators can be expanded as a sum over bosonic fieldoperators a†,a:

Φ†(x, y) =∑

i a†iW

∗i (x, y) . (355)

70

The usual single-band Bose-Hubbard model [1, 27] is obtained via the tight binding and lowest band approxima-tions [85]. A Gaussian of width σ is used for the lowest band Wannier basis:

Wi(x, y) = exp−[(x− xi)2 + (y − yi)

2]/4σ2/√

2πσ , (356)

where (xi, yi) are the coordinates of the ith site. The rotational part HL ≡∫

d2r Φ†ΩLzΦ of Eq. (353) becomes

HL = −i~Ω∫

d2r∑

<i,j>[

a†j ai(W∗j (x, y)(x∂y − y∂x)Wi(x, y)) + h.c.

]

≡ i~Ω∑

<i,j>Kij(aia†j − a†i aj) , (357)

where <i, j > indicates a sum over nearest neighbors. To obtain Kij , we take σ = d/2 and evaluate the integral foran infinite system [99]. Then

Kij = (rirj/ed2) sinαij , (358)

where ri denotes the distance from the axis of rotation to the ith site and αij is the angle subtended by the ith andthe jth sites with respect to the axis of rotation. Note that Eq. (358) is a purely geometrical, dimensionless factor.

Thus, the rotating Bose-Hubbard Hamiltonian is

H = − t∑

<i,j>(aia†j + a†i aj)

− i~Ω∑

<i,j>Kij(aia†j − a†i aj)

+ 12U∑

i ni(ni − 1) − µ∑

i ni , (359)

in the grand canonical ensemble. The hopping t and on-site interaction U have been explicitly calculated from g, m,etc. elsewhere [85]. Equation (359) is the usual Bose-Hubbard Hamiltonian with the addition of the second term,which describes the rotating frame.

B. Physical Measurable Quantities: Observables

The expectation value of the current is an important observable in our analysis. In the Heisenberg picture thecurrent is given by the Heisenberg equation of motion:

〈Jij〉 = (i/~d)〈[ni, Hij ]〉 (360)

=it

~d〈aia†j − a†i aj〉 −

ΩKij

d〈aia†j + a†i aj〉 ,

where Hij is the Hamiltonian for sites i, j alone. In Eq. (360), the first term is due to hopping while the second is due

to current. Note that the sum over all nearest neighbors j of 〈Jij〉 is zero, so that current is conserved in the rotatingframe. Another useful observable is the total current on the lattice boundary C,

Λ ≡ (~d/Er)∑

〈i,j〉∈C〈Jij〉 , (361)

where we have scaled away the units via a “recoil” energy Er ≡ ~2/Md2 and all sums over C are taken with thesame sign convention for helicity as Ω. We also define two number-related observables: n =

∑

i〈ni〉, the average total

number of atoms in the system, and ν ≡ ∑i(〈ni2〉 − 〈ni〉2)/∑

i〈ni〉, the normalized variance. Recall that ν = 1 fora coherent state, ν > 1 for a phase-squeezed state, ν < 1 for a number-squeezed state, and ν = 0 for a single Fockstate.

For a Fock space consisting of only zero or one atoms per site, phase differences can be defined consistently. Supposethe ground state has one atom. Number conservation leads to a wavefunction of form

|ψ〉 = c1|1102 · · · 0N〉 + c2|0112 · · · 0N 〉+...+ cN |0102 · · · 1N〉 , (362)

where subscripts within the kets are site indices and c1, . . . , cN are complex numbers, with N the total number ofsites. Then the total phase winding on C is

θ ≡∑〈i,j〉∈C θij , θij ≡ [arg(ci) − arg(cj)] , (363)

71

1 2 32

3

4

5

6

nΛ

U=0.5U=2.0U=3.0

U=10.0

U=1000

FIG. 22: (color online) The total dimensionless current Λ around a lattice unit cell is shown as a function of the average totalnumber of atoms n in the system. The curves from top to bottom are for increasing on-site interaction strength U in units of thelattice recoil energy Er. For weak interactions, the atoms behave independently; for strong interactions, there is a particle-holesymmetry, as evident in the bottommost curve, where n = 1 and n = 3 have the same Λ. The curves here are a guide to theeye.

which is 2π times an integer value.One can develop an algorithmic generalization of the definition of θ in a consistent way for up to N total atoms

in the system. For simplicity, we will focus our discussion on the strongly interacting case, achieved experimentallyeither by a Feshbach resonance or by turning up the lattice potential height to be very high, which reduces thehopping. In this case one can make a two-state approximation [3] which prevents multiple occupancy of the samesite: this requires 2N kets as a basis of the many body system. Such a Fock space eliminates the interaction term Uin Eq. (353). However, there are effective strong interactions due to atoms being unable to cross each other; i.e., onehas a system of hard core bosons.

C. Test Cases

Having defined the observables, we first consider the case of a single square unit cell, i.e., a 2 × 2-site lattice. Inorder to assess the two-state approximation, we allow Fock states with up to three atoms per site: however, wekeep the average number of particles per site at unity or below. We exactly diagonalize the Hamiltonian and findthe ground state. Due to the competition between the hopping and rotational energy terms in Eq. (359), rotationenters the system only when ~Ω > t. For an average total number of atoms n ∈ 0, 1, 2, 3, θ = 0 for ~Ω < t andθ = 2π for ~Ω > t. For non-interacting atoms (U = 0), this corresponds to Λ = 0 and Λ = 2n, respectively. As theinteratomic repulsion is increased, Λ decreases to non-integer values, as shown in Fig. 22. Although θ is quantized,Λ is not. However, for t/U ≪ 1 and ~Ω/U ≪ 1, i.e., for very strong interactions, and/or a very strong lattice andsmall rotation, the allowed values of Λ return to those given by the two-state approximation. Thus, the two-stateapproximation is adequate to study strongly interacting systems.

Consider next a 4 × 4-site square 2D lattice, which consists of 9 unit cells. This is sufficient to describe a vortexlattice, as we will demonstrate. The two-state approximation is computationally necessary for exact diagonalization,so we study the strongly interacting case: a priori this is a 216-ket basis. First, consider the simplified case of oneatom in the system. Number conservation reduces the ground state to the simple form of Eq. (362). Numericalstudy of this case produces two main results. (1) Second order quantum phase transitions occur each time a unitof total phase winding θ enters the system. (2) The maximum θ is 6π. For higher fillings, Both the hopping t andthe rotation Ω(directional hopping) can drive the system through the extensively studied Mott-insulator/superfluidtransition [1, 27, 86, 87].

To illustrate result (1), in Fig. 23 is shown the total phase winding and the derivative of the total energy E ≡ 〈H〉with respect to Ω. Energy level crossings (not shown) are also observed, corresponding to each transition. Thecorresponding circulation patterns are shown in Fig. 24. In Fig. 24(a), θ = 0 and rotation has not yet entered thesystem. In the rotating frame, the current seems to be flowing backwards, i.e., clockwise. In Fig. 24(b), θ = 2π anda single vortex enters the system and rests at the center. Figure 24(c) shows one of two possible patterns for θ = 4π.Figure 24(d) is a tightly packed vortex lattice; the unit of negative vorticity at the center is a trivial result of packingfour vortices together on a 2D square lattice.

72

0 0.2 0.4 0.6 0.8 101234

Ω

m0 0.2 0.4 0.6 0.8 1

0

10

20

ΩL

(a)

(b)

FIG. 23: (color online) (a) Phase winding around a 4 × 4 lattice as a function of the rotation Ω in units of Er/~, with totalnumber of atoms n = 1 and hopping t/Er = 1. (b) The derivative of the ground state energy with respect to Ω.

b b

FIG. 24: (color online) The red arrows indicate direction of current flow for t/Er = 1 and n = 1 at increasing rotations on a4 × 4 lattice,corresponding to phase windings of θ = 0, 2π, 4π, 6π. (a) ~Ω/Er = 0.5; (b) ~Ω/Er = 1.0; (c) ~Ω/Er = 2.0; (d)~Ω/Er = 4.0. The last case is a vortex lattice.

Result (2) follows from the point symmetry group Cj , which describes the rotation of a j-fold symmetric object,e.g., a j-sided regular polygon. The number of possible rotational symmetries is equal to j. Thus the lattice unit celldetermines the number of such phases: for the square lattice there are four, characterized by phase windings aroundthe border of 0, 2π, 4π, and 6π; for the hexagonal lattice there are six, and so forth.

As in the case of the unit cell (see Fig. 22), Λ depends on the number of atoms and the strength of the interactionbetween them; it also depends on θ. This latter dependence can be understood as follows. For one atom in the system,from Eqs. (360)-(362),

Λ =∑

C 2|cicj |(t sin θij −Kij~Ω cos θij)/Er . (364)

Since θij increases in discrete steps, Λ depends linearly on Ω with a different slope for each phase winding. For θ = 6πand 16 sites on C, θij = π/2 and the slope is zero. If one normalizes Eq. (364) to the total number of atoms on theboundary nC ≡∑i∈C〈ni〉, the maximum value of Λ is exactly 2t/Er.

D. General Filling: The Many Body Problem

Finally, we extend our results to the case of general filling. The main difference from the single-atom case is thepresence of effective strong interactions due to the use of the two-state approximation. In Fig. 25 are illustrated thenumber density nd, normalized variance ν, total phase winding θ, and the normalized current on the boundary Λ/nC ,all for quarter filling. Note that nC depends on Ω. In Fig. 25(a)-(b) the site-dependence of nd and ν are illustrated for~Ω = t = Er. In general, the density at the center varies for the same winding number, so that it is not an indicationof vortex core size; in this strongly interacting system the core size is smaller than d. In Fig. 25(c) one observes thatΛ depends linearly on Ω between quantum phase transitions; this is qualitatively a similar result to that of one atom,

73

1 2 3 41

23

4

00.20.4

1 2 3 41

23

4

00.5

1

(a) Number density (b) Normalizedvariance ν

0 0.2 0.4 0.6 0.8 1−2

−1

0

1

2

3

Ω

Λ C /nC an

d m

FIG. 25: (color online) Observables for a 4 × 4 lattice with quarter-filling and t/Er = 1. (a) Site-dependent number densityn for ~Ω/Er = 1. There is noticeable depletion on the inner sites. (b) Site-dependent normalized variance ν for ~Ω/Er = 1.Note that ν < 1, so this is a number-squeezed ground state. (c) The normalized, scaled current Λ/nC (solid curve) and thetotal phase winding (dashed curve), all on the boundary as a function of ω(in units of Er/~). Between each quantum phasetransition, the normalized current depends linearly on the rotation Ω.

as seen in Eq. (364). However, max(Λ/nC) 6= 2, due to interactions. The other essential features of general filling, i.e.,the occurrence of quantum phase transitions and energy level crossings as units of phase winding enter the system,are qualitatively the same as that for one atom. This is a non-trivial result, as, in contrast to a single unit cell or asingle atom on a 4 × 4 lattice, general filling is not analytically tractable.

Finite-size effects make it difficult to use plots such as Fig. 23(b) to conclusively establish quantum phase transitionsfor a 4 × 4 lattice. However, based on symmetry considerations, we identify transitions between rotational groundstates as second order quantum phase transitions. As Ω increases, the state of the system changes continuously whilethe phase winding, a symmetry-linked property, changes discontinuously. This fits with the definition of a second orderphase transition [28]. Further, the system adopts higher rotational symmetries (1-fold, 2-fold and 4-fold) for higherrotational energy. This resembles the adoption of higher symmetry states at higher temperature seen in classicalsecond order phase transitions. Finally, there is a small discrete set of total phase windings; this set is independentof the system size. Identical rotational constraints point to the same set of total phase windings for the rotating 3Dlattice although the exact nature of transitions will depend on dimensionality.

We note that the appearance of a vortex lattice in the discrete system should be observable by expansion andinterference with a non-rotating system, as in the continuous case [88]. A second observable has been provided by ourcalculations: the current on the boundary jumps discontinuously with increasing Ω. The maximal value of Λ at totalphase winding θ = 6π, indicates a tightly packed vortex lattice.

IX. NUMERICAL METHOD III: TIME EVOLVING BLOCK DECIMATION

A. Quantum Many-Body Theory

We now turn to the full quantum many-body description of the problem in the regime of optical lattice depthswhere the above mean-field tight-binding description is appropriate, and beyond. As was demonstrated by Jakschet al. [86], a system of weakly interacting ultracold bosons loaded in an optical lattice potential is an almost perfectrealization of the Bose-Hubbard Hamiltonian (BHH), a model introduced to the condensed matter community almostten years earlier by Fisher et al. [27].

To derive the BHH, we start with the 1D continuous many-body Hamiltonian in second quantization for two-bodyinteractions:

H =

∫

dx؆(x)

[

− ~2

2m

∂2

∂x2+ Vext(x)

]

Ψ(x)

+1

2

∫

dx

∫

dx′Ψ†(x)Ψ†(x′)Vint(x − x′)Ψ(x′)Ψ(x) . (365)

We then expand the bosonic field operator Ψ(x), which destroys a particle at position x, in a lowest Bloch band

Wannier basis as Ψ(x) =∑

i biw(x − xi), where the operator bi is defined to destroy a particle in the localizedWannier wave function w(x − xi). This step is analogous to expansion of the full condensate wave function in alocalized basis when discretizing the continuous NLS to obtain the DNLS. Then, as is done in the derivation of thecontinuous NLS, we assume the two-body interaction potential to be of the contact form, i.e., Vint(x−xi) = gδ(x−xi),where g is proportional to the s-wave scattering length of the atoms. We also invoke the tight-binding approximationby assuming that the lattice is deep enough to obtain sufficiently localized Wannier functions. This allows us to discard

74

all terms except those involving nearest-neighbor hopping and on-site interactions. After making these assumptions,we arrive at the familiar BHH:

H = −JM−1∑

i=1

(b†i+1bi + h.c.) +U

2

M∑

i=1

ni(ni − 1) +

M∑

i=1

ǫini, (366)

where bi and b†i are destruction and creation operators at site i that obey the usual bosonic commutation relations,

and ni ≡ b†i bi is the number operator which counts the number of bosons at site i. Equation (366) assumes boxboundary conditions on a lattice containing M sites. The coefficients J , U , and ǫi can be calculated exactly in termsof the localized single-particle wave functions and other parameters [86]. J is the nearest-neighbor hopping coefficient,U is the on-site interaction energy, and ǫi is an external potential. The ratio U/J is an important parameter thatdetermines the relative contribution from each term. For a shallow lattice, the hopping term dominates, whereas fora deep lattice the interaction term dominates. However, we note that U is not completely dependent on the latticegeometry since the s-wave scattering length can be varied independently via a Feshbach resonance.

In deriving the BHH, one makes very similar assumptions to those made when discretizing the continuous NLS ona lattice to obtain the DNLS. Specifically, both derivations invoke a lowest Bloch band tight-binding approximation.However, in the latter case, a single configuration of bosons is assumed from the onset. The full quantum treatmentallows for quantum depletion out of the condensate mode and thus can describe the system in the strongly interactingregime.

A general pure state of the full many-body quantum system can be written as a complex linear superposition ofstates, each with a well-defined number of particles in each Wannier state:

|Ψ〉 =

d−1∑

n1,n2,...,nM=0

cn1n2···nM |n1n2 · · ·nM 〉, (367)

where nk is the number of particles at site k. For obvious computational reasons, we truncate the local Hilbert spaceat local dimension d, i.e., we restrict the occupation of each Wannier state to contain at most d−1 bosons. The Hilbertspace containing all pure states of the full many-body system is thus of dimension dM which becomes prohibitivelylarge for large systems. For example, even for d = 2 we can only simulate M = 12 or 13 lattice sites on a singlePC without further refining the numerical algorithm. We overcome this difficulty with use of the time-evolving blockdecimation routine which will be discussed in Sec IX C.

B. Discrete Mean-Field Theory From Discrete Quantum Many-Body Theory

Next, we show how the DNLS can be recovered from the BHH. The destruction operator at site k can be evolved

in time in the Heisenberg picture according to i~∂tbk = [bk, H ]. After computing the commutators, we arrive at

i~∂tbk = −J(bk+1 + bk−1) + Ubkb†kbk + ǫk bk. (368)

We can then take the expectation value of Eq. (368) to obtain an equation of motion for the order parameter 〈bk〉.The DNLS is recovered exactly if the expectation value is taken with respect to a product of atom-number Glaubercoherent states. That is, for full many-body states of the form

|Ψ〉 =M⊗

k=1

|zk〉, where |zk〉 = e−|zk|2

2

∞∑

n=0

znk√n!|n〉, (369)

we obtain the DNLS for the equation of motion governing the coherent state amplitude zk = 〈bk〉:

i~∂tzk = −J (zk+1 + zk−1) + U |zk|2zk + ǫkzk. (370)

As discussed in Sec. IXA, for numerical calculations we must truncate the local Hilbert space to a finite dimensiond, in which case the on-site coherent states of Eq. (369) become truncated coherent states:

|Ψ〉 =M⊗

k=1

|zk〉, where |zk〉 = Nd e− |zk|2

2

d−1∑

n=0

znk√n!|n〉, (371)

75

and Nd is a normalization factor.The coherent states of Eq. (369) are known to well-describe the ground state of the BHH for J ≫ U in the limit of

an infinite number of sites M and particles N at fixed filling N/M [89]. It is in this regime that quantum depletioncan be safely neglected and Eq. (370) is an accurate description of the system. However, the lattice must still be deepenough so that the single-band tight-binding approximation is still valid. In Sec. IXE, we use the truncated coherentstates of Eq. (371) to create nonequilibrium initial quantum states in the BHH that are analogs to the dark solitonsolutions of the DNLS.

C. Time-Evolving Block Decimation Algorithm

The time-evolving block decimation (TEBD) algorithm was first introduced in 2003-2004 by Vidal [46, 90] inthe context of quantum computation. Soon thereafter, Daley et al. [91] and White and Feiguin [92] translated thealgorithm into more the familiar density matrix renormalization group (DMRG) language and showed that TEBD isequivalent to a time-adaptive DMRG routine. Here, we summarize our implementation of TEBD as applied to theBHH.

1. The Vidal Decomposition

The Vidal prescription is to first rewrite the coefficients in Eq. (367) as a product of M tensors Γ[ℓ] and M − 1vectors λ[ℓ]:

cn1n2···nM =

χ∑

α1,...,αM−1=1

Γ[1]n1

α1λ[1]α1

Γ[2]n2

α1α2λ[2]α2

Γ[3]n3

α2α3· · ·Γ[M ]nM

αM−1. (372)

There does exist a procedure for determination of the Γs and λs given known coefficients of an arbitrary state; however,this is not generally useful because one does not typically have access to each component of the dM -dimensional vector|Ψ〉. This procedure would require a Schmidt decomposition (SD) at every bipartite splitting of the lattice, whereχ ≤ d⌊M/2⌋ is the number of Schmidt vectors retained at each splitting. The Schmidt number χS , i.e., the numberof Schmidt basis sets required for an exact representation of the state at each cut, is naturally a measure of globalentanglement between the lattice sites [90, 93]. The decomposition (372) is thus appropriate when |Ψ〉 is only slightlyentangled according to the Schmidt number, in which case it is computationally feasible to take χ ≈ χS .

2. Two-Site Operation

One of the reasons why this decomposition is useful is that it allows for efficient application of two-site unitaryoperations. Let us consider a two-site unitary operation V =

∑

Vnℓnℓ+1

n′ℓn

′ℓ+1

|nℓnℓ+1〉〈n′ℓn

′ℓ+1| acting on sites ℓ and ℓ+ 1.

First, we write |Ψ〉 in terms of Schmidt vectors for the subsystems [1 · · · ℓ− 1] and [ℓ+ 2 · · ·M ]:

|Ψ〉 =∑

αℓ−1,αℓ,αℓ+1;nℓ,nℓ+1

λ[ℓ−1]αℓ−1

Γ[ℓ]nℓαℓ−1αℓ

λ[ℓ]αℓ

Γ[ℓ+1]nℓ+1αℓαℓ+1

|Φ[1···ℓ−1]αℓ−1

〉 ⊗ |nℓnℓ+1〉 ⊗ |Φ[ℓ+2···M ]αℓ+1

〉

=∑

αℓ−1,αℓ+1;nℓ,nℓ+1

Θnℓnℓ+1αℓ−1αℓ+1

|Φ[1···ℓ−1]αℓ−1

〉 ⊗ |nℓnℓ+1〉 ⊗ |Φ[ℓ+2···M ]αℓ+1

〉 (373)

by invoking Eqs. (13) and (14) of [90], where

Θnℓnℓ+1αℓ−1αℓ+1

≡∑

αℓ

λ[ℓ−1]αℓ−1

Γ[ℓ]nℓαℓ−1αℓ

λ[ℓ]αℓ

Γ[ℓ+1]nℓ+1αℓαℓ+1

λ[ℓ+1]αℓ+1

(374)

and αℓ ∈ 1, 2, . . . , χ. Note that this definition of the tensor Θ differs from an analogous construct in [90] which isalso denoted Θ in that work. We are up to this point assuming that we know the decomposition (372) of |Ψ〉, andhence we also know all elements of Θ. However, by writing |Ψ〉 in the form of Eq. (373) we can easily write the

updated state after the application of V as

V |Ψ〉 =∑

αℓ−1,αℓ+1;nℓ,nℓ+1

Θnℓnℓ+1αℓ−1αℓ+1

|Φ[1···ℓ−1]αℓ−1

〉 ⊗ |nℓnℓ+1〉 ⊗ |Φ[ℓ+2···M ]αℓ+1

〉, (375)

76

where Θ can be written in terms of the updated tensors Γ[ℓ] and Γ[ℓ+1] and the updated vector λ[ℓ]:

Θnℓnℓ+1αℓ−1αℓ+1

=∑

n′ℓ,n

′ℓ+1

Vnℓnℓ+1

n′ℓn

′ℓ+1

Θnℓnℓ+1αℓ−1αℓ+1

=∑

αℓ

λ[ℓ−1]αℓ−1

Γ[ℓ]nℓ

αℓ−1αℓλ

[ℓ]αℓ

Γ[ℓ+1]nℓ+1

αℓαℓ+1λ[ℓ+1]αℓ+1

. (376)

In practice, a given two-site operation is performed as follows: (1) form Θ from current Γs and λs [Eq. (374)]; (2)

update Θ by applying V to obtain Θ [Eq. (376)]; (3) reshape Θ from a 4-tensor to a (χd)× (χd) matrix; (4) perform

a singular value decomposition (SVD) on this matrix retaining only the largest χ singular values λ[ℓ]αℓ ; and (5) divide

out the previous values of λ[ℓ−1] and λ[ℓ+1] in order to compute Γ[ℓ] and Γ[ℓ+1] from the matrix elements obtained viathe SVD. The most expensive computational steps are (1), the formation of Θ, and (2), the update of Θ after the

application of V . The former requires O(d2χ3) elementary operations, whereas the latter requires O(d4χ2) elementaryoperations; hence, our overall two-site operation scales as O[max(d2χ3, d4χ2)].

3. Real Time Evolution

The BHH is a sum of one- and two-site operations, but the terms multiplying J in Eq. (366) do not all commute,

so the time evolution operator e−iHt/~ does not directly factor into a product of one- and two-site unitary operations.However, because the BHH only links nearest neighbors, we write H = Hodd + Heven, where

Hodd = − J∑

i odd

(b†i+1bi + h.c.) +∑

i odd

[

U

2ni(ni − 1) + ǫini

]

and (377)

Heven = − J∑

i even

(b†i+1bi + h.c.) +∑

i even

[

U

2ni(ni − 1) + ǫini

]

. (378)

Each term within both Hodd and Heven commute even though [Hodd, Heven] 6= 0. It is then convenient to utilizea Suzuki-Trotter approximation of the time evolution operator for small time steps δt. Specifically, we employ

the second-order expansion: e−iHδt/~ ≈ e−iHoddδt/2~e−iHevenδt/~e−iHoddδt/2~, where each exponential factor can befactored into a product of two-site unitaries. Even though the terms involving n are one-site operations, we stilltreat them as two-site operations by appropriate tensor products with the identity operator. In practice, we buildd2-dimensional matrix representations of H for each lattice link and diagonalize these matrices to obtain matrixrepresentations of the two-site unitary operators. Then, in conjunction with the Suzuki-Trotter expansion, we employthe two-site operation procedure outlined in Sec. IXC 2 on an initial decomposed configuration |Ψ〉 O(M) times foreach of tf/δt total time steps, updating the decomposition at each step. It is straightforward to calculate single-site observables, e.g., the expectation value of the number operator 〈nk〉, and two-site observables, e.g., the one-

body density matrix 〈b†i bj〉, by using the partial trace to calculate the reduced density matrix of the subsystem of

interest. For example, to calculate 〈b†i bj〉, we first compute ρij = Trk 6=i,j |Ψ〉〈Ψ| using Eq. (372) for |Ψ〉 and then use

〈b†i bj〉 = Tr(b† ⊗ b ρij). Overall, our implementation of the TEBD algorithm scales as O[Mtfδt max(d2χ3, d4χ2)].

4. Sources of Error and Convergence Properties

The TEBD algorithm makes two important approximations: (1) the retention of only the χ most heavily weightedbasis sets during a given two-site operation (see Sec. IXC 2), and (2) the Suzuki-Trotter representation of the timeevolution operator (see Sec. IXC 3). For the latter case, we find that for the results presented in Sec. IXE it issufficient to use time steps of size δt = 0.01 ~/J to obtain converged results. The former approximation is more subtleas its accuracy is directly related to the amount of entanglement present in the system. Specifically, in Sec. IXE, wetime evolve mean-field initial states [see Eq. (371)] for which χ = 1 is sufficient for exact representation; however,unitary time evolution increases entanglement between sites. To ensure that our choice of χ is sufficient, we runequivalent simulations with increasing values of χ and look for convergence of calculated observables, e.g., averagelocal number 〈nk〉. It is important to point out that, owing to the local nature of the expansion (372), accuratecalculation of nonlocal observables, e.g., off-diagonal elements of the single-particle density matrix, converge moreslowly with respect to χ. For the observables and time scales presented in Sec. IXE, our results are converged for thespecified values of χ = 45, 50. We also note that the fidelity of truncation at χ eigenvalues can be quantified by thesum of the non-retained eigenvalues after application of a two-site operation. This quantity can be interpreted as ameasure of the amount of entanglement generated by the two-site operation. Typically, for a single two-site unitary,we find this truncation error to be less than or on the order of 10−6.

77

D. Constrained Imaginary Time Relaxation in DNLS

1. Fundamental Dark Soliton Solutions

In Sec. IXE, we use the TEBD routine to simulate the quantum evolution of the dark soliton solutions of theDNLS by using truncated coherent states as initial configurations. This requires knowledge of the set of coherentstate amplitudes zk corresponding to a discrete dark soliton. Using a standard Crank-Nicolson scheme for the time-stepping procedure, we calculate the standing dark soliton solution of the DNLS by performing constrained imaginarytime relaxation on Eq. (370) with ǫk = 0. Specifically, we take an initial condition of form zk = mxk where xk is the

position of the kth site and x = 0 is the center of the lattice and normalize the solution to NDNLS =∑Mk=1 |zk|2 at

each step of imaginary time. The stationarity of the solution is tested by subsequent evolution in real time.

2. Density and Phase Engineering of Gray Solitons

We also consider the case of two solitons moving toward one another at finite velocity. These initial conditions areobtained via the methods of density and phase engineering for soliton creation [12] as applied to the DNLS. We firstperform imaginary time relaxation on a uniform initial condition with an external potential of the form

ǫk = V0

exp

[

− (xk + ξ)2

2σ2ǫ

]

+ exp

[

− (xk − ξ)2

2σ2ǫ

]

(379)

to dig two density notches each centered at distance ξ from the center of the lattice. Next, we imprint an instantaneousphase of the form

θk = ∆θ

−1

2tanh

[

2(xk + ξ)

σθ

]

+1

2tanh

[

2(xk − ξ)

σθ

]

+ 1

(380)

which gives the solitons equal-and-opposite initial velocities toward the center of the lattice. Phonon generation isminimized by appropriately tuning the width σθ of the phase profiles to the soliton depth as determined by V0 in thedensity engineering stage.

E. Time Evolution of Quantum Solitons

With an initial DNLS configuration zk(0) obtained either via the procedure outlined in Sec. IX E for a singlestanding soliton or the procedure outlined in Sec. IXD 2 for two colliding solitons, we then build a product oftruncated coherent states according to Eq. (371) for input into the TEBD quantum simulation routine. The Vidal

decomposition (372) of a product state |Ψ〉 =⊗M

k=1

(

∑d−1nk=0 c

(k)nk |nk〉

)

is trivial to compute:

λ[ℓ]αℓ

= δαℓ,1 and Γ[ℓ]nℓαℓ−1αℓ

= c(ℓ)nℓδαℓ−1,1δαℓ,1, (381)

where for the case of truncated coherent states c(k)n = Nd e

−|zk|2/2 znk√n!

.

An extensive discussion of results obtained using the above methodology to create nonequilibrium dark solitoninitial states in the BHH is presented in Refs. [94, 95]. We will summarize those results here. Being direct analogsof mean-field solitons, the initial conditions analyzed below do not conserve total particle number, although quantumevolution does conserve total average particle number. That is, in this section, we consider the quantum many-bodyevolution of mean-field-like solitons and refer to these structures as quantum solitons. However, we stress that neitherthe discrete mean-field theory (DNLS) nor the corresponding quantum theory (BHH) are integrable systems, so theseare not solitons in the mathematically rigorous sense. It is possible to density and phase engineer dark soliton statesdirectly in the BHH that are eigenfunctions of the total number operator. In Ref. [95], we use a number-conservingversion of the TEBD routine to generate and analyze the quantum dynamics of such states. Although some observablesbehave differently in this case, the conclusions reached are generally the same.

1. Standing Solitons

The DNLS assumes a single configuration of bosons, i.e., bosons are only allowed to occupy one single-particleorbital. However, for an M mode system, a full quantum treatment will permit bosons to occupy any of the M

78

〈nk〉

tJ/h

Site Index k

(a)

−10 0 100

5

10

15

20

25

0.2

0.4

0.6

0.8

1

1.2

N0|φ0k|2

tJ/h

Site Index k

(b)

−10 0 100

5

10

15

20

25

0.2

0.4

0.6

0.8

1

1.2

|〈bk〉|2

tJ/h

Site Index k

(c)

−10 0 100

5

10

15

20

25

0.2

0.4

0.6

0.8

1

1.2

FIG. 26: Density measures for a standing quantum soliton. Quantum evolution of (a) average particle number, (b) condensatewave function [96], and (c) order parameter versus position and time for a standing DNLS dark soliton initial configuration.The dashed line in (a) indicates the 1/e decay time of the order parameter norm Nb.

permissible modes. For the system sizes accessible to explore with the TEBD routine, a standing soliton initial DNLSconfiguration exhibits a finite lifetime due to quantum effects indescribable by mean-field theory. Most notably,quantum evolution causes significant quantum depletion out of the initial dark soliton configuration into higher orderorbitals which fill in the soliton density notch, where as discussed in [94], the natural orbitals of the system are defined

as the eigenfunctions of the one-body density matrix 〈b†i bj〉 [96]. We find that the soliton lifetime is closely correlated

to the the growth in quantum effects such as a decay in the order parameter norm Nb =∑Mk=1 |〈bk〉|2 and growth in

the generalized entropy Q = dd−1

[

1 − 1M

∑Mk=1 Tr(ρ2

k)]

. Shown in Fig. 26 is the evolution of density measures of a

quantum soliton with parameters νU/J = 0.35, ν = 1, M = 21, d = 7, χ = 45, where ν = NDNLS/M is approximatelyequal to the average filling.

2. Soliton-Soliton Collisions

In Fig. 27, we display the quantum evolution of two colliding dark solitons where the initial conditions wereobtained by density and phase engineering in the DNLS as summarized in Sec. IXD 2. Here, if the decoherence time,as measured by the decay time of the order parameter norm, occurs before or near the collision time, then there is aloss in elasticity of the soliton collision. For a fixed value of the effective nonlinearity νU/J , we can independently tunethe decoherence time by changing the filling ν without altering the initial density-phase profile of the solitons. In Fig.27, we depict this effect in three separate simulations with parameters νU/J = 0.35, M = 31, χ = 50, V0/J = 0.4,σǫ/a = 1, ∆θ = 0.3π, σθ/a = 2, ξ/a = 6 at filling factors ν = 1, 0.5, 0.1, where a is the lattice constant.

〈nk〉

tJ/h

Site Index k

(a)

−10 0 100

5

10

15

20

0.4

0.6

0.8

1

1.2

〈nk〉

tJ/h

Site Index k

(b)

−10 0 100

5

10

15

20

0.2

0.3

0.4

0.5

0.6

〈nk〉

tJ/h

Site Index k

(c)

−10 0 100

5

10

15

20

0.04

0.06

0.08

0.1

0.12

FIG. 27: Quantum soliton collisions and decoherence-induced inelasticity. Average particle number for two colliding quantumsolitons at filling factors (a) ν = 1, (b) ν = 0.5, and (c) ν = 0.1. The collision elasticity is decreased when the decoherencetime (dashed lines) occurs at or before the time of collision.

79

X. APPENDIX: SIMPLE EXERCISES TO LEARN EXACT DIAGONALIZATION

The following is a simple set of exercises to work through numerical solutions of the Bose-Hubbard model in a FockState Basis. I include an elementary discussion of the Hamiltonian, to aid those who feel lost with the more advanceddiscussion in the main body of the course. A set of progressive exercises follow. Solutions are available upon request.

A. Bose-Hubbard Model

The Bose-Hubbard model describes a system that consists of cold bosons in a one-dimensional optical lattice. ForBose-Einstein condensates, the 1-D lattice is usually produced experimentally from two counter-propagating laserbeams along the x-axis together with counter-propagating laser beams of much higher intensity in the other twospatial dimensions, i.e., the y- and z-directions. Note that the model applies equally well to both 2-D and 3-D opticallattices.

The Bose-Hubbard Hamiltonian can be derived from first principles using quantum field theory and applying twoapproximations: (1) the particles only occupy the lowest energy level at each site in the lattice, and (2) the “tight-binding approximation” is valid, which essentially assumes that the atom density is highly confined into the minimum

of each site in the lattice. At this point is easiest to just think of the system as a given number of potential wellscontaining atoms that can move around and interact with one another. The form of the Bose-Hubbard Hamiltonianthat I use (you may see others in the literature) is given as follows:

H = −J∑

〈i,j〉(b†i bj + bib

†j) +

U

2

∑

i

ni(ni − I) + V∑

〈i,j〉ninj − µ

∑

i

ni . (382)

The first term describes classically forbidden hopping (tunneling) between nearest-neighbor sites; the second termaccounts for interactions between particles that exist at the same lattice site; the third term accounts for interactionsbetween particles at nearest-neighbor lattice sites (is usually small and is neglected in most models); and the final

term accounts for the chemical potential of the atoms. bi and b†i are operators which destroy and create a particle,respectively, at site i. The number operator ni can be written in terms of the destruction and creation operators:

ni = b†i bi. The 〈i, j〉 notation under the summations indicates a sum over nearest neighbors. The parameters J , U ,V , and µ are controllable experimentally and determine the strengths of the respective terms. For instance, small Jand large U can be obtained by turning up the height of the lattice so that the particles cannot tunnel through thebarrier to an adjacent lattice site.

The construction of the terms is very intuitive. There are two terms present in the tunneling term to allow particlesto hop both in the left and right directions. The particles gain an energy equal to J when they hop from one site toanother. The factor ni(ni − 1)/2 in the second term is simply the number of on-site interactions: every particle ata given site interacts with every particle but itself, hence the ni(ni − 1); the 2 in the denominator out front avoidsovercounting the number of interactions by a factor of two. Using combinatorics, we note that ni(ni − 1)/2 =

(

ni

2

)

issimply the number of ways of “choosing” two interacting particles from the occupation ni. U thus denotes the energyper interaction. A similar reason could be applied to the nearest-neighbor interaction term. A good discussion ofthese concepts is given in [97].

B. Our Hilbert Space: Fock Space

The state space of the problem is known as a Fock space or number space. The basis states in this space have

well-defined occupation, i.e., number of particles, at each lattice site. These basis states are commonly known as Fockstates or number states. Technically, the number states for the full many-body (i.e., many-well) quantum system aretensor products of number states at each lattice site:

|Ψ〉Fock =

N⊗

i=1

|ni〉 = |n1〉 ⊗ |n2〉 ⊗ · · · ⊗ |nN 〉, (383)

where N is the total number of lattice sites and ni indicates the number of particles at site i. To simplify the notation,(383) is often written as

|Ψ〉Fock = |n1 n2 · · · nN 〉 = |M〉, (384)

80

where the M just indicates that we are talking about the Mth number state.Of course, quantum mechanically we can have states other than the number states, but since the number states

form a complete, orthogonal set, we can write any state as a linear superposition of the number states:

|Ψ〉 =∑

j

Cj |j〉, (385)

where the states |j〉 are number states as in (384), and Cj = 〈j|Ψ〉 is the probability amplitude that our generic state|Ψ〉 is found in the pure number state |j〉.

The destruction, creation, and number operators mentioned above in Sec. XA behave exactly as those encounteredin the quantum harmonic oscillator problem (which you should review) except that we now have multiple lattice sitesand the operators destroy/create particles rather than quanta of energy:

bi|n1 n2 · · · ni · · · nN 〉 =√ni|n1 n2 · · · ni − 1 · · · nN 〉,

b†i |n1 n2 · · · ni · · · nN 〉 =√ni + 1|n1 n2 · · · ni + 1 · · · nN 〉, (386)

ni|n1 n2 · · · ni · · · nN 〉 = ni|n1 n2 · · · ni · · · nN 〉.

C. Generating Matrix Representations of Operators

All of these operators can be represented as matrices in our number space (unless the space is total particle number

conserving, as we will see). A generic operator A has elements defined as follows:

Aij = 〈i|A|j〉, (387)

where the states |i〉 and |j〉 are base states in an arbitrary basis (the number basis in our case). The reasoning behind

Eq. (387) can be understood by examining the expectation value of the operator A in a given state |Ψ〉 by projectingthe state |Ψ〉 onto the |i〉 basis:

A = 〈Ψ|A|Ψ〉 =

(

∑

i

〈Ψ|i〉〈i|)

A

∑

j

|j〉〈j|Ψ〉

=∑

i,j

〈Ψ|i〉〈i|A|j〉〈j|Ψ〉. (388)

We can recognize the terms in the double summation as a vector element 〈Ψ|i〉 times a matrix element 〈i|A|j〉 times

another vector element 〈j|Ψ〉, hence the matrix representation of A given by (387). By the way, the reason 〈j|Ψ〉represents a vector element can be understood if one writes |Ψ〉 as a superposition of base states |k〉:

〈j|Ψ〉 = 〈j|(

∑

k

Ck|k〉)

=∑

k

Ck〈j|k〉 = Cj , (389)

where the last equality follows because the states |j〉 and |k〉 are orthogonal: 〈j|k〉 = δj,k.Note that our full many-body system Fock states are orthogonal in this way too. For example, the inner product

〈2 3 0 1|1 2 0 3〉 = 0. Don’t be tempted to compute this as one would an ordinary dot product in a Cartesian spacetreating each occupation as the length of the vector in each “dimension”: 〈2 3 0 1|1 2 0 3〉 6= 2 · 1 + 3 · 2 + 0 · 0 + 1 · 3.(A mistake both Dimitri and myself made at the onset of our projects.)

D. Exercises

1. Suppose we have two wells and one total particle in the system. In the Fock basis our two base states are thus|11 02〉 and |01 12〉. In vector notation, these can be written as

|11 02〉 =

(

1

0

)

and |01 12〉 =

(

0

1

)

, (390)

where we could have just as well assigned the |11 02〉 state to the second unit vector in the Hilbert space and|01 12〉 to the first. For clarity, I have put a subscript below the occupation number in the kets indicating

81

lattice site number. Note that we have not specified a local dimensions of bosons (as Carr did in the meeting):this is a particle-conserving number space.

Try computing the Hamiltonian [Eq. (382)] for this system with V = 0 (the V term would have no ef-fect anyhow since we only have one total particle) by using Eq. (387). Note that since we don’t allow morethan one particle at any lattice site at the same time, the on-site interaction term in the Hamiltonian alsohas no effect. Because this is a particle-conserving state space, the destruction and creation operators do notexist in this space. (We cannot create/destroy particles in a total particle number conserving system without

destroying/creating them somewhere else.) The product b†i bj , however, does have a matrix representation inthis space since the particle that gets destroyed at site j is simultaneously created at site i. You must thus

generate matrix representations of the operators b†i bj and then add them up to get the full tunneling termof the Hamiltonian. Remember, the matrix representation of any operator is given by (387), and the effectof the destruction, creation, and number operators is defined in (386). In our model, the indices i and jin the tunneling term will only differ by one since we only allow nearest-neighbor tunneling. Don’t let thenearest-neighbor summation notation in (382) confuse you: for a one-dimensional system j = i+1 and the sum-mation runs from site i = 1 to site i = N−1 (why not all the way to N?), with N the total number of lattice sites.

With the Hamiltonian computed, solve for the eigenvalues/eigenstates of the Hamiltonian for in termsof J and µ. This is a simple eigensystem calculation as is done in Math Phys. and Linear Algebra. Physically,the eigenstates of the Hamiltonian are states of constant probability density, and they govern the time-evolutionof the system.

2. Now still with one total particle, extend the above calculations to three wells. Your Hilbert space will now bethree-dimensional. Again compute the eigensystem of the Hamiltonian in terms of J and µ (go ahead and useMathematica).

3. Now consider a system composed of a single well. We allow 0, 1, or 2 particles in the well, so that your threestates are now |01〉, |11〉, and |21〉. With only a single site, you can ignore the tunneling and nearest-neighborinteraction terms altogether. Compute the Hamiltonian for this system. The Hamiltonian should already bediagonal since there is no tunneling to link adjacent states, hence there is no need to diagonalize it. Thus, thenumber states are the stationary states of the system (a rather trivial result) with eigenvalues given by thediagonal elements of the Hamiltonian.

4. Our system now consists of two wells, and we specify a local dimension d = 2. Physically, such a local dimensionrepresents strongly interacting bosons: two bosons aren’t allowed to occupy the same site at the same time asit is energetically unfavorable. Our dN = 4 number states are thus

|0〉 = |01 02〉, |1〉 = |01 12〉, |2〉 = |11 02〉, |3〉 = |11 12〉, (391)

where the |j〉, j ∈ 0, 1, 2, 3, are simply unit vectors in the full many-body four-dimensional Hilbert space.Physically, the ordering of the number states above, i.e., which state I called |0〉, |1〉, etc., is irrelevant. However,it is algorithmically important to select an appropriate ordering scheme. The ordering in (391) applied to largersystems turns out to be very convenient. Also, because we are no longer conserving total number of particles inthe state space, it is now legal to compute the matrix representations of the destruction and creation operatorsseparately and then multiply them together when computing the Hamiltonian. This turns out to be a relativelyeasy way to compute the Hamiltonian for larger systems.

5. Compute the matrix-form of the Hamiltonian for a number-conserving system of two wells and two particles.“Number-conserving“ implies that a state of |0〉 = |0102〉 can never be connected to a state of |1〉 = |0112〉, i.e.you cannot create a particle from nothing and you cannot destroy a particle completely.

Now, compute the Hamiltonian for a number-conserving system of two wells and a local dimension, d = 3 (upto two particles per site).

6. Generalize the number-conserving Hamiltonian to N particles on M sites. Then generalize the number-conserving Hamiltonian to a local dimension of d, for N particles on M sites.

Make a program for the full generalization using Matlab, Mathematica, or your programming language of choice.For this exercise, set M = 2, and set the local dimension d = N + 1. Typical values for the input parametersare V = 0, U = 2, J = 1, and µ = 1. Make plots of |cnL |2 (the square of the eigenvectors) as a function of k(index) and nL (number of particles in well 1). Plot the spectra of Ek (eigenvalues) with respect to nL. Playwith the ratio of J to U . What happens when U = 0? Do this for several different values of N .

82

[1] S. Sachdev, Quantum Phase Transitions (Cambridge University Press, New York, 1999).[2] A. Lewenstein, M. Sanpera, V. Ahufinger, B. Damski, A. Sen De, and U. Sen, Adv. Phys. 56, 243 (2007).[3] L. D. Carr and M. J. Holland, Phys. Rev. A 72, 033622(R) (2005).[4] R. Kanamoto, L. D. Carr, and M. Ueda, Phys. Rev. Lett. 100, 060401 (2008).[5] M. L. Wall and L. D. Carr, New Journal of Physics in press, e-print http://arxiv.org/abs/0812.1548 (2009).[6] A. L. Fetter and A. A. Svidzinsky, J. Phys.: Condens. Matter 13, R135 (2001).[7] Emergent Nonlinear Phenomena in Bose-Einstein Condensates, edited by P. G. Kevrekidis, D. J. Frantzeskakis, and R.

Carretero-Gonzalez (Springer-Verlag, Berlin, 2008).[8] L. D. Carr, C. W. Clark, and W. P. Reinhardt, Phys. Rev. A 62, 063610 (2000).[9] L. D. Carr, C. W. Clark, and W. P. Reinhardt, Phys. Rev. A 62, 063611 (2000).

[10] L. D. Carr, M. A. Leung, and W. P. Reinhardt, J. Phys. B: At. Mol. Opt. Phys. 33, 3983 (2000).[11] L. D. Carr, J. N. Kutz, and W. P. Reinhardt, Phys. Rev. E 63, 066604 (2001).[12] L. D. Carr, J. Brand, S. Burger, and A. Sanpera, Phys. Rev. A 63, 051601 (2001).[13] L. Khaykovich, F. Schreck, F. Ferrari, T. Bourdel, J. Cubizolles, L. D. Carr, Y. Castin, and C. Salomon, Science 296, 1290

(2002).[14] L. D. Carr and J. Brand, Phys. Rev. Lett. 92, 040401 (2004).[15] L. D. Carr and C. W. Clark, Phys. Rev. Lett. 97, 010403 (2006).[16] http://alps.comp-phys.org.[17] A. F. Albuquerque, F. Alet, P. Corboz, P. Dayal, A. Feiguin, S. Fuchs, L. Gamper, E. Gull, S. Gurtler, A. Honecker, R.

Igarashi, M. Korner, A. Kozhevnikov, A. Lauchli, S. R. Manmana, M. Matsumoto, I. P. McCulloch, F. Michel, R. M.Noack, G. Pawlowski, L. Pollet, T. Pruschke, U. Schollwock, S. Todo, S. Trebst, M. Troyer, P. Werner, and S. Wessel,Journal of Magnetism and Magnetic Materials 310, 1187 (2007).

[18] R. Bhat, M. J. Holland, and L. D. Carr, Phys. Rev. Lett. 96, 060405 (2005).[19] R. V. Mishmash and L. D. Carr, Journal of Mathematics and Computers in Simulation in press, e-print

http://arxiv.org/abs/0810.2593 (2008).[20] R. Shankar, Principles of Quantum Mechanics (Springer, ADDRESS, 1994).[21] L. E. Ballentine, Quantum Mechanics: A Modern Development (World Scientific Publishing, Hackensack, NJ, 1998).[22] K. Gottfried and T.-M. Yan, Quantum Mechanics, 2nd ed. (Springer, New York, 2004).[23] M. Le Bellac, Quantum Physics (Cambridge University Press, New York, 2006).[24] A. M. L. Messiah and O. W. Greenberg, Phys. Rev. 136, B248 (1964).[25] J. Bros, A. M. L. Messiah, and D. N. Williams, Phys. Rev. 149, 1008 (1966).[26] N. N. Boguliubov, J. Phys. 11, 23 (1947).[27] M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D. S. Fisher, Phys. Rev. B 40, 546 (1989).[28] L. D. Landau and E. M. Lifshitz, Statistical Physics (Pergamon Press, New York, NY, 1969), Vol. 5.[29] R. L. Stratonovich, Sov. Phys. Dokl. 2, 416 (1958).[30] J. Hubbard, Phys. Rev. Lett. 3, 77 (1959).[31] F. Zhou, e-print cond-mat/0505740 (2005).[32] F. Zhou and C. Wu, New J. Phys. 8, 166 (2006).[33] D. B. M. Dickerscheid, D. van Oosten, E. J. Tillema, and H. T. C. Stoof, Phys. Rev. Lett. 94, 230404 (2005).[34] L.-M. Duan, Phys. Rev. Lett. 95, 243202 (2005).[35] B. DeMarco and D. S. Jin, Science 285, 1703 (1999).[36] M. Kohl, H. Moritz, T. Stoferle, K. Gunter, and T. Esslinger, Phys. Rev. Lett. 94, 080403 (2005).[37] C. A. Regal, M. Greiner, and D. S. Jin, Phys. Rev. Lett. 92, 040403 (2004).[38] M. Greiner, C. A. Regal, and D. S. Jin, e-print cond-mat/0407381 (2004).[39] In the opposite limit, Jf ≫ Vf , the fermionic field is superfluid except for even integer filling. The coupling introduces

intersite correlations into the bosonic field, and thus destroys the boson Mott state.[40] V. J. Emery, Phys. Rev. B 14, 2989 (1976).[41] For a filling factor of n = 0, 2, the Mott state is identical with a band insulator state, even for Jf ≫ Vf .[42] A. Auerbach, Interacting Electrons and Quantum Magnetism (Springer, Berlin, 1994).[43] An extension of the two-state approximation to superpositions between |nb〉 and |nb +1〉 obtains the correct Bose-Hubbard

superfluid-Mott borders [1, 27] for arbitrary nb.[44] E. Timmermans, Phys. Rev. Lett. 87, 240403 (2001).[45] T. D. Kuhner and H. Monien, Phys. Rev. B 58, R14741 (1998).[46] G. Vidal, Phys. Rev. Lett. 93, 040502 (2004).[47] G. K. Brennen, Quant. Inf. Comp. 3, 619 (2003).[48] H. Barnum, E. Knill, G. Ortiz, and L. Viola, Phys. Rev. A 68, 032308 (2003).[49] H. Barnum, E. Knill, G. Ortiz, R. Somma, and L. Viola, Phys. Rev. Lett. 92, 107902 (2004).[50] T. D. Kuhner, S. R. White, and H. Monien, Phys. Rev. B 61, 12474 (2000).[51] D. Gobert, C. Kollath, U. Schollwock, and G. Schutz, Phys. Rev. E 71, 036102 (2005).[52] J. Kevorkian and J. Cole, Multiple Scale and Singular Perturbation Methods, 1st edition ed. (Springer, New York, 1996).[53] A. H. Nayfeh, Perturbation Methods (Wiley-Interscience, ADDRESS, July 12, 2000), Vol. 1st edition.

83

[54] L. P. Pitaevskii, Sov. Phys. JETP 13, 451 (1961).[55] E. P. Gross, Nuovo Cimento 20, 454 (1961).[56] C. Sulem and P. L. Sulem, Nonlinear Schrodinger Equations: Self-focusing Instability and Wave Collapse (Springer-Verlag,

New York, 1999).[57] E. Madelung, Z. Phys. 40, 322 (1926).[58] L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Addison-Wesley, Massachusets, 1959).[59] A. L. Fetter and J. D. Walecka, Theoretical Mechanics of Particles and Continua (McGraw-Hill, Inc., New York, 1980).[60] P. G. Saffman, Vortex Dynamics (Cambridge Univ. Press, New York, 1992).[61] A. L. Fetter, Phys. Rev. 138, A429 (1965).[62] P. A. Ruprecht, M. J. Holland, K. Burnett, and M. Edwards, Phys. Rev. A 51, 4704 (1995).[63] Y. S. Kivshar, Physics Reports 298, 81 (1998).[64] V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 34, 62 (1972).[65] V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 37, 823 (1973).[66] R. M. Miura, J. Math. Phys. 9, 1202 (1968).[67] R. M. Miura, S. C. Gardner, and M. D. Kruskal, J. Math. Phys. 9, 1204 (1968).[68] V. M. Perez-Garcıa, H. Michinel, and H. Herrero, Phys. Rev. A 57, 3837 (1998).[69] F. Bowman, Introduction to Elliptic Functions, with Applications (Dover, New York, 1961).[70] Handbook of Mathematical Functions, edited by M. Abramowitz and I. A. Stegun (National Bureau of Standards, Wash-

ington, D. C., 1964).[71] W. P. Reinhardt and C. W. Clark, J. Phys. B: At. Mol. Opt. Phys. 30, L785 (1997).[72] S. Burger, K. Bongs, S. Dettmer, W. Ertmer, K. Sengstock, A. Sanpera, G. V. Shlyapnikov, and M. Lewenstein, Phys.

Rev. Lett. 83, 5198 (1999).[73] J. Denschlag, J. E. Simsarian, D. L. Feder, C. W. Clark, L. A. Collins, J. Cubizolles, L. Deng, E. W. Hagley, K. Helmerson,

W. P. Reinhardt, S. L. Rolston, B. I. Schneider, and W. D. Phillips, Science 287, 97 (2000).[74] J. N. Elgin, Phys. Rev. A 47, 4331 (1993).[75] L. Berge, T. J. Alexander, and Y. S. Kivshar, 1999, e-print cond-mat/9907408.[76] C. A. Jones and P. H. Roberts, J. Phys. A: Math. Gen. 15, 2599 (1982).[77] F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, Rev. Mod. Phys. 71, 463 (1999).[78] W. Ketterle, D. S. Durfee, and D. M. Stamper-Kurn, in Proceedings of the International School of Physics “Enrico Fermi”

(IOS Press, Amsterdam; Washington, D.C., 1999), pp. 67–176.[79] H. Michinel, V. Perez-Garcıa, and R. de la Fuente, Phys. Rev. A 60, 1513 (1999).[80] J. D. Close and W. Zhang, J. Opt. B 1, 420 (1999).[81] L. Dobrek, M. Gajda, M. Lewenstein, K. Sengstock, G. Birkl, and W. Ertmer, Phys. Rev. A 60, R3381 (1999).[82] P. W. Anderson, Basic Notions of Condensed Matter Physics (Addison-Wesley, New York, 1984).[83] C. Wu, H. Chen, J. Hu, and S. C. Zhang, Phys. Rev. A 69, 043609 (2004).[84] R. Bhat, P. M. Peden, B. T. Seaman, M. Kramer, L. D. Carr, and M. J. Holland, Phys. Rev. A 74, 063606 (2006).[85] A. M. Rey, K. Burnett, R. Roth, M. Edwards, C. J. Williams, and C. W. Clark, J. Phys. B: At. Mol. Opt. 36, 825 (2003).[86] D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, Phys. Rev. Lett. 81, 3108 (1998).[87] M. Greiner, O. Mandel, T. Esslinger, T. W. Hansch, and I. Bloch, Nature 415, 39 (2002).[88] K. W. Madison, F. Chevy, W. Wohlleben, and J. Dalibard, Phys. Rev. Lett. 84, 806 (2000).[89] W. Zwerger, Journal of Optics B: Quantum and Semiclassical Optics 5, S9 (2003).[90] G. Vidal, Phys. Rev. Lett. 91, 147902 (2003).[91] A. J. Daley, C. Kollath, U. Schollwock, and G. Vidal, J. Stat. Mech. - Theor. Exp. 2004, P04005 (2004).[92] S. R. White and A. E. Feiguin, Phys. Rev. Lett. 93, 076401 (2004).[93] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cam-

bridge, UK, 2000).[94] R. V. Mishmash and L. D. Carr, e-print arXiv:0710.0045 (2007).[95] R. V. Mishmash, I. Danshita, C. W. Clark, and L. D. Carr, Phys. Rev. A, to be submitted (2009).[96] O. Penrose and L. Onsager, Phys. Rev. 104, 576 (1956).[97] D. Jaksch and P. Zoller, Annals of Physics (2004).[98] Recall that the Young’s modulus is the stiffness of an isotropic elastic material, F = Y ∆L

Lin 1D.

[99] For smaller lattices, the error due to the finite size introduces a small correction to the constant e in Eq. (358).