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Quantum Statistical Mechanics

Path integrals for bosons, fermions, and quantumspin systems

Alejandro Muramatsu

Institut fur Theoretische Physik IIIUniversitat Stuttgart

Les Houches Doctoral Training in Statistical Physics

2 A. Muramatsu - Quantum Statistical Mechanics

Contents

1 Path integrals for bosons 51.1 Quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Quantum field-theory in d-dimensions . . . . . . . . . . . . . . . . . . 91.3 Coherent states for bosons . . . . . . . . . . . . . . . . . . . . . . . . 101.4 Path-integrals for bosons . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Path integrals for fermions 172.1 Grassmann algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2 Coherent states for fermions . . . . . . . . . . . . . . . . . . . . . . . 20

2.2.1 Gaussian integrals . . . . . . . . . . . . . . . . . . . . . . . . 222.3 Path-integrals for fermions . . . . . . . . . . . . . . . . . . . . . . . . 23

3 Path integrals for spins 273.1 Coherent states for spins . . . . . . . . . . . . . . . . . . . . . . . . . 273.2 Path-integrals for quantum spin systems . . . . . . . . . . . . . . . . 31

3.2.1 Berry phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4 Topological excitations in quantum spin systems 394.1 The non-linear σ-model . . . . . . . . . . . . . . . . . . . . . . . . . . 394.2 Topological excitations . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.2.1 The XY-model and vortices . . . . . . . . . . . . . . . . . . . 414.2.2 The O(3) non-linear σ-model and skyrmions . . . . . . . . . . 444.2.3 Anyons in 2+1 dimensions . . . . . . . . . . . . . . . . . . . . 49

3

Chapter 1

Path integrals for bosons

There is an intimate connection between statistical mechanics and quantum field-theory that is best seen by discussing the path-integral formulation of quantummechanics. We are not going to dwell much on the historical reasons or in a detaileddiscussion of the equivalences but concentrate more on those aspects that are closerto our initial motivation, that is to statistical mechanics. Such a connection is mostuseful when discussing the critical properties of many-body quantum systems, e.g.close to a quantum phase transition, where the wealth of knowledge from statisticalmechanics can be used to describe the universality class to which a given quantumcritical point belongs. We will also see in later chapters, that specific properties ofquantum systems rely on the topological properties of certain terms that may arise,whose geometric interpretation becomes clear in a path integral description.

We start these lectures by considering first the simplest case of a single degree offreedom and its path-integral, as introduced by Feynman [1]. Then, this chapter willbe dedicated to bosons, i.e. degrees of freedom that have a classical limit. Startingfrom a Hamiltonian in second quantization we will arrive by means of a coherent-state representation at the corresponding path-integral.

1.1 Quantum mechanics

We discuss first the path-integral formulation for a single particle as was initiallyintroduced by Feynman. As in the elementary lectures in quantum mechanics, westart by considering the Hamiltonian of the system

H =p2

2m+ V (q) , (1.1)

where p and q are momentum and position operators, respectively, m is the massof a particle and V (q) a potential. Such a Hamiltonian was obtained in canonicalquantization by looking at the Hamilton-function of the classical system and pro-moting the canonical variables p and q into operators. Since we expect that at somepoint the action appears, let us recall its form. The Hamilton function H(p, q) and

5


the Lagrangian L(q, q) are connected by a Legendre transformation

H = pq − L , (1.2)

such that the Lagrangian is given by

L =1

2mq2 − V (q) , (1.3)

so that the action is

S =

∫dt L(q, q, t) =

∫dt

[1

2mq2 − V (q)

]. (1.4)

We will see now that starting from the Hamiltonian (1.1), we will arrive to anexpression containing the action (1.4) but fully in the frame of quantum mechanics.

Let us calculate the amplitude for the system to reach the state |qf > at time t,if it was in state |qi > at time t = 0. This amplitude can be expressed by the timeevolution operator

< qf |e−i~Ht|qi >

=< qf |e−iH(t−tN−1)/~ · · · e−iH(tj+1−tj)/~ · · · e−iHt1/~|qi > , (1.5)

where we have devided the time interval t in N segments of equal length that wedenote from now on ε ≡ tj+1 − tj. Between each pair of evolution operators forintervals ε, we can introduce the identity 1 =

∫dqj|qj >< qj|, such that

< qf |e−i~Ht|qi >

=

∫ N−1∏i

dqi < qf |e−iεH/~|qN−1 > · · · < q1|e−iεH/~|qi > . (1.6)

However, since H contains both operators p and q that do not commute, we splitthe kinetic and potential terms making an error of O (ε2), that in the limit N 1is small. The splitting above is known as Trotter-Suzuki slicing [2, 3], and is oftenused in quantum Monte Carlo algorithms (see e.g. [4]).

e−iεH/~ = exp

(−i ε

~p2

2m

)exp

[−i ε

~V (q)

]+O

(ε2). (1.7)

Here we used the Baker-Campbell-Hausdorff formula

eε(A+B) = eεAeεBeε2C , (1.8)

where

C =1

2[A,B] +O(ε) . (1.9)

A. Muramatsu - Quantum Statistical Mechanics 7

The commutator [A,B] leads in our case to

[A,B] → − 1

2m~2

[p2, V (q)

]. (1.10)

Assuming that V (q) is a smooth function of q, we can perform a Taylor expansionsuch that [

p2, V (q)]→ [p, qn] = −i~nqn−1 . (1.11)

Since q is self-adjoint, eε2C is a unitary operator that in the limit ε → 0 converges

to the identiy. In this way we have on the one hand

exp[−i ε

~V (q)

]| qj > = e−iεV (qj)/~ | qj > . (1.12)

On the other hand, for the kinetic term we have to go over to eigenstates of p

|q >=

∫dp√2π~

eipq/~|p > , (1.13)

such that for an infinitesimal time interval we have

< qj+1| exp

(−i ε

~p2

2m

)|qj >

=

∫dpj+1√

2π~

∫dpj√2π~

e−i(pj+1qj+1−pjqj)/~ exp

(−i ε

~p2j

2m

)< pj+1|pj >︸︷︷︸δ(pj+1−pj)

= limδ→0+

1

2π~

∫dpj e

−ipj(qj+1−qj)/~ exp

[−i(ε− iδ)

~p2j

2m

]=

√m

2πi~εexp

[i

~m

2ε(qj+1 − qj)2

]. (1.14)

Putting everything together, and taking the limit N → ∞, we have for thetransition amplitude,

< qf |e−i~Ht|qi >

= limN→∞

[ m

2πi~ε

]N/2 ∫ N−1∏i

dqi

× exp

i

~

N−1∑j=0

[m2ε

(qj+1 − qj)2 − εV (qj)]

, (1.15)

with the boundary conditions q0 = qi and qN = qf . Since in the limit N →∞, ε→ 0,∑j ε is a Riemann summation, and hence, asuming continuity of the variables, we

can go over to an integral. In the same way, the kinetic term gives a derivative

limε→0

1

ε(qj+1 − qj)2 = lim

ε→0ε

(qj+1 − qj

ε

)2

= ε

(dqjdt

)2

. (1.16)


Finally we can write the transition amplitude in the form of a path integral

< qf |e−i~Ht|qi > = C

∫Dq(t) eiS/~ , (1.17)

where the action is given by

S =

∫ t

0

dt

[1

2m

(dq

dt

)2

− V (q)

], (1.18)

and C takes into account all the multiplicative constants entering (1.15). We alsointroduced the notation ∫

Dq(t) =

∫ N−1∏i

dqi , (1.19)

denoting the integration over all possible trajectories statrting at qi at t = 0, andending at qf at time t.

Several remarks are pertinent here.

i) The quantum mechanical amplitude for a transition can be obtained on thebasis of the classical action, where not only the classical path satisfying δS = 0is present but all possible paths are included. In this representation we do notdeal with operators anymore.

ii) The classical limit is obtained in a natural way. If we take the limit ~ → 0,the saddle-point gives the only contribution, that corresponds to δS = 0.

iii) In the derivation above we assumed continuity of the dynamical variables. Itis not a priori clear that this is true.

It is also interesting to discuss the possibility of taking imaginary times τ = it.This is nothing else than an analytic continuation of (1.17) into imaginary time, alsocalled a Wick rotation. In doing so, we have

< qf |e−1~Hτ |qi > = C ′

∫Dq(τ) e−SE/~ , (1.20)

with the euclidean action

SE =

∫ τ

0

dτ

[1

2m

(dq

dτ

)2

+ V (q)

], (1.21)

and a real constant C ′. Notice that the action contains an inverted potential (V →−V ). The interest on imaginary times has several reasons:

i) The expression (1.20) is purely real, and hence amenable to a direct compu-tation, where each classical path has a positive definite weight exp (−SE/~).This renders the problem into a purely statistical one.


ii) In the case we would like to compute the partition function of a quantummechanical problem, we have

Z = Tr e−βH , (1.22)

such that we need to consider the evolution of the states entering the tracealong the imaginary time ~β.

iii) If we are interested only in the ground-state | ΨG > of a system, we can projectit out of a trial wave function | ΨT >, as long as < ΨT | ΨG >6= 0, since

limθ→∞

e−θH | ΨT >∼| ΨG > . (1.23)

Also here we can view the projection as an evolution in imaginary time.

1.2 Quantum field-theory in d-dimensions

Without trying to be rigorous, let us construct by analogy the partition function fora system described by fields. Starting with a Hamiltonian in d dimensions of thefollowing form:

H =

∫ddr

[1

2Π2(r) +

1

2(∇φ)2 + V (φ)

], (1.24)

where canonical conjugated fields satisfying [φ (x) ,Π (x′)] = iδ (x− x′), are as-sumed (from now on ~ = 1).

The partition function in imaginary time should have the form

Z =

∫Dφ e−SE , (1.25)

with the action

SE =

∫ddr

∫dτ

[1

2φ2 +

1

2(∇φ)2 + V (φ)

]=

∫dd+1r

[1

2(∇φ)2 + V (φ)

]. (1.26)

The expressions above should emphasize the analogy between statistical mechanicsin (d + 1) dimensions and euclidean quantum field-theory in d dimensions, whereimaginary time becomes an extra dimension for the corresponding field-theory instatistical mechanics. Below we tabulated a dictionary translating between objectsin statistical mechanics and quantum field-theory.

Statistical Mechanics Quantum Field-TheoryHamiltonian Euclidean action

Partition function Z Generating functionalCorrelation function Expectation value of time ordered products

correlation length ∼ ξ → e−rξ mass → e−mr

free energy F = − lnZ Ground-state energy(T − TMF

c ) ∼ m20 bare mass


1.3 Coherent states for bosons

Here we will follow the discussion by Negele and Orland [5]. We start by consideringa system of bosons with pairwise interactions, where the Hamiltonian is given by

H =∑i,j

Tij b†ibj +

1

2

∑i,j,k,`

Vi,j,k,` b†ib†jbkb` , (1.27)

where b†i and bi are creation and annihilation operators acting on state |i〉, respec-tively, obeying bosonic commutation relations[

bi, b†j

]= δij ,

[bi, bj

]=[b†i , b

†j

]= 0 . (1.28)

Now we recall that for the construction of the Feynman path-integral, an im-portant step was to perform a Trotter splitting, where between the time-slices acomplete set of states for the operators p or q was introduced. Since in the presentcase we have operators in second quantization, by analogy, we seek for eigenstatesof bosonic creation and/or annihilation operators. The set of possible states belongto the Fock space, where a general state contains all possible occupation numbers:

|φ〉 =∞∑n=0

∑α1,...,αn

φα1,...,αn|α1, . . . , αn〉 , (1.29)

with the case n = 0 corresponding to the vacuum. We consider first the possibilityof |φ〉 being an eigenstate of a creation operator b†α. Since |φ〉 has a component witha minimum number of particles, if we create a particle, we increase the minimumnumber by one, such that the new state cannot be proportional to the old one, andhence, cannot be an eigenstate. The second possiblity is |φ〉 being an eigenstate ofan annihilation operator bα. If we apply it to (1.29), we could in principle have astate proportional to |φ〉, since all possible number of particles are present.

A convenient basis for the Fock-space is given by occupation number states, sincethey are already symmetrized and orthonormalized. In this representation, a generalstate can be expanded as

|φ〉 =∑

nα1 ,...,nαj ,...

φnα1 ,...,nαj ,...|nα1 , . . . , nαj , . . . 〉 . (1.30)

If |φ〉 is an eigenstate of an annihilation operator, it should fulfil

bαi |φ〉 = φαi |φ〉 . (1.31)

Since |nα1 , . . . , nαj , . . . 〉 constitute a basis, the eigenvalue equation leads to the con-dition

φαiφnα1 ,...,nαi−1,... =√nαiφnα1 ,...,nαi ,...

. (1.32)


This relation leads to a recursion for φnα1 ,...,nαi ,...until we reach nαi = 0. By carrying

out such a recursion for all the states αi, we arrive at

φnα1 ,...,nαi ,...=

φnα1α1√nα1 !· · · φ

nαiαi√nαi !· · · . (1.33)

On the other hand, the occupation number states can be expressed as

|nα1 , . . . , nαi , . . . 〉 =b†nα1α1√nα1 !· · ·

b†nαiαi√nαi !· · · |0〉 , (1.34)

such that together with (1.33) leads to the following form for an eigenstate of anni-hilation operators

|φ〉 =∏α

∑nα

(φαb

†α

)nαnα!

|0〉 = exp

(∑α

φαb†α

)|0〉 . (1.35)

Once we have explicitely constructed a coherent state, we can see that its adjointis a left eigenstate of the creation operator:

〈φ|b†α = 〈φ|φ∗α . (1.36)

Furthermore, the following relations hold:

b†α|φ〉 =∂

∂φα|φ〉 ,

〈φ|bα =∂

∂φ∗α〈φ| ,

〈φ|φ′〉 = exp

(∑α

φ∗αφ′α

). (1.37)

An important property we need for setting up a path integral is that the set ofstates we use is complete. For coherent states one speaks of the resolution of unity.It is accomplished by the following form:∫ ∏

α

dφ∗αdφα2πi

exp

(−∑α

φ∗αφα

)|φ〉〈φ| = 1 , (1.38)

where 1 is the identity operator in Fock-space. We can show that the above is trueby directly performing the integration. Going over to the real and imaginary partsof φα, we have for the measure

dφ∗αdφα2πi

=dReφαdImφα

π, (1.39)


such that using (1.33) we obtain∫ ∏α

dφ∗αdφα2πi

exp

(−∑α

φ∗αφα

)|φ〉〈φ|

=∑

nα1 ,...,nαi ,...

n′α1,...,n′αi ,...

|nα1 , . . . , nαi , . . . 〉〈n′α1, . . . , n′αi , . . . |

×∏αi

∫dρidθiπ

ρie−ρ2i

(ρie

iθi)nαi√

nαi !

(ρie−iθi)n′αi√

n′αi !

=∑

nα1 ,...,nαi ,...

|nα1 , . . . , nαi , . . . 〉〈nα1 , . . . , nαi , . . . | = 1 . (1.40)

Since as shown in (1.37) the coherent states are in general not orthogonal, the setof coherent states is overcomplete.

1.4 Path-integrals for bosons

Now we have all the ingredients to set up the path-integral for the Hamiltonian(1.27). To be specific, and since we are focusing on statistical mechanics, we considerthe partition function in the grand-canonical ensemble.

Z = Tr e−βK

= C

∫ ∏α

dφ∗αdφα exp

(−∑α

φ∗αφα

)〈φ|e−βK |φ〉 , (1.41)

where we defined the operator

K ≡ H − µN , (1.42)

as a short notation. As in the case of the Feynman path-integral we introduce aTrotter slicing

e−βK =L∏`=1

e−∆τK , (1.43)

where ∆τ = β/L. Then, the resolution of unity is introduced between the time-slicesand we take a normal ordered form

e−∆τK = : e−∆τK : +O(∆τ 2

), (1.44)

where normal ordering means that all the creation operators are at the left of allannihilation operators. Here we have an error of O (∆τ 2) that will vanish in the


limit L→∞. The partition function (1.41) looks now as follows:

Z ∝ limL→∞

∫ L∏`=1

∏α

dφ∗α,`dφα,` exp

(−

L∑`=1

∑α

φ∗α,`φα,`

)

×L∏`=1

〈φ`|1−∆τK +O(∆τ 2

)|φ`−1〉 , (1.45)

where we used the convention that |φ0〉 = |φL〉, and φα,L = φα,0 = φα. The matrix

elements of K(b†α, bβ

)between coherent states are easily obtained, since those states

are left and right eigenstates of b†α and bα, respectively.

〈φ`|K(b†α, bβ

)|φ`−1〉 = K

(φ∗α, φβ

)〈φ`|φ`−1〉 , (1.46)

where now K is a complex valued function of φ∗α, φα. Recalling that the overlapbetween coherent states is given in (1.37), we have finally,

Z = limL→∞

∫ L∏`=1

∏α

dφ∗α,` dφα,` e−S(φ∗,φ) , (1.47)

where

S(φ∗, φ) =L∑`=2

∆τ

∑α

φ∗α,`

[(φα,` − φα,`−1)

∆τ− µφα,`−1

]+H

(φ∗α,`, φα,`−1

)

+∆τ∑α

φ∗α,1

[(φα,1 − φα,L)

∆τ− µφα,L

]+H

(φ∗α,1, φα,M

). (1.48)

As will be seen in the exercises, such a discrete form is the one that should be usedfor actual calculations. However, the usual notation, that makes a connection witha field theory, is assuming continuity of the fields, resulting in

Z =

∫Dφ∗α(τ)Dφα(τ) e−S , (1.49)

where the action is given by

S =

∫ β

0

dτ∑α

[φ∗α(τ)

(∂

∂τ− µ

)φα(τ) +H (φ∗α(τ), φα(τ))

]. (1.50)

We also introduced the notation∫Dφ∗α(τ)Dφα(τ) = lim

L→∞

∫ L∏`=1

∏α

dφ∗α,` dφα,` , (1.51)

in a similar way as done for the Feynman path-integral. Due to the fact thatthe expression above originates from a trace, the fields obey periodic boundaryconditions in imaginary time: φ(β) = φ(0).


In the course of the calculation above we have discarded a multiplicative constantin the partition function. The reason is that a physical system will be completelydescribed by its correlation functions in equilibrium. They can be obtained by at-taching infinitesimal external fields and considering the response of the system totheir variations. Moreover, we are generally interested in the cumulants, that is thecorrelations between fluctuations. For example, if we are interested in some observ-able O(x), we may consider its expectation value 〈O(x)〉 (one point function) orthe response of the system to a field that couples to O in the form of a susceptibility(two point function)

χ(x,x′) = 〈O(x)O(x′)〉 − 〈O(x)〉〈O(x′)〉 , (1.52)

that tells us about the fluctuations and their correlations in the system. In generalwe could wish to calculate an N -point function. This can be easily achieved byadding a source term to the action of the form

S → S +

∫dτ ddx

∑α

[J∗α(x, τ)φα(x, τ) + φ∗α(x, τ)Jα(x, τ)] . (1.53)

Then, for e.g. a two point correlator for bosons we have

〈bα(x, τ)b†β(x′, τ ′)〉 =δ2 lnZ

δJ∗α(x, τ)δJβ(x′, τ ′)

∣∣∣∣∣J=J∗=0

. (1.54)

Since for cumulants we need to have the logarithm of the partition function, mul-tiplicative constants do not contribute to the result, and hence, can be ignored atthe outset. From the discussion above, we see that the partition function, with theinclusion of the appropriate source terms, can be used to obtain any desired corre-lation function. Therefore, it is called the generating functional, as we pointed outat the end of Sec. 1.2.

Problem: Partition function for the harmonic oscillator

Consider the Lagrangian

L(x, x) =1

2mx2 − 1

2mω2x2 . (1.55)

i) Give the form of the euclidean action using the expression (1.15), introducingthe imaginary time τ = it, and setting N∆τ = ~β.

ii) Give explicitly the form of the partition function, showing that x(τj) = x(τj +~β).


iii) Due to the periodic boundary conditions of x(τj), it can Fourier-transformed:

x(τj) =1√~β

∑ω

xω e−iτj ω . (1.56)

Show that ω can only take discrete values ωn with n ∈ Z, and show whichthey are.

iv) Give the form of the action in terms of xn.

v) Give the form of the partition function up to a normalization constant comingfrom the Fourier-transformation. Here it is convenient express it in terms ofn ≥ 0.

vi) Carry out the integrals. Since x0 is real, it is convenient to treat it separately.

vii) Show that the result obtained can be expressed as a product of the resultfor ω = 0, i.e. the case of a free particle and a rest. Disregard the partcorresponding to the free case, lumping it into the normalization constant.

viii) Using that

∞∏n=1

(1 +

x2

n2

)=

sinhπx

πx, (1.57)

display the partition function in such a way that the eigenvalues of the har-monic oscillator are obtained.

Chapter 2

Path integrals for fermions

The main difficulty encountered in the case of fermions is the lack of a classical limitfor them, in the sense that C-numbers cannot reflect the anticommuting character offermions. We will motivate the introduction of anticommuting objects by followingthe same ideas we used in contructing coherent states for bosons. As in the case ofbosons we follow mainly the discussion by Negele and Orland [5].

2.1 Grassmann algebras

We start, by essentially following the same path we had in the case of bosons, andconsider a system of fermions with pairwise interactions, where the Hamiltonian isgiven by

H =∑i,j

Tij f†i fj +

1

2

∑i,j,k,`

Vi,j,k,` f†i f†j f`fk , (2.1)

where f †i and fi are creation and annihilation operators acting on state |i〉, respec-tively, obeying fermionic anticommutation relations

fi, f†j

= δij ,

fi, fj

=f †i , f

†j

= 0 . (2.2)

In the bosonic case we looked for eigenstates of the annihilation operator, thatturned out to be coherent states. Let us assume that such a state exists in our Fockspace of interest. Then, it should fulfil

fα|φ〉 = φa|φ〉 . (2.3)

We can now consider another annihilation operator, and assuming that the eigen-values are C-numbers, we have

fαfβ|φ〉 = φβφα|φ〉 , (2.4)

while for the other ordering we have

fβfα|φ〉 = φαφβ|φ〉 , (2.5)

17


but due to the anticommutation relations (2.2), the eigenvalues have also to an-ticommute, i.e. it should hold that φβφα = −φαφβ. Therefore, they cannot beC-numbers.

We therefore consider now a set of elements ξα, α = 1, . . . , n with the property

ξαξβ + ξβξα = 0 , (2.6)

i.e. anticommuting objects, such that ξ2α = 0. They are the generators of a Grass-

mann algebra, that may be defined on R or on C. We recall that an algebra is a setof elements with addition and product as possible operations. The generators giveall the elements of the algebra by their products and linear combinations thereof.In the following we list a number of properties that will turn out to be necessary infinding the fermionic coherent states.

• Given n generators, we can construct a basis of the algebra by all possibleproducts of the form

ξm11 ξm2

2 · · · ξmnn , with mi = 0, 1 , (2.7)

such that it contains 2n elements: 1, ξα1 , ξα1ξα2 , . . . , ξα1ξα2 · · · ξαn. A generalelement of the algebra is given by a linear combination of the elements of thebasis.

• If n is even, we define the conjugation by assigning to each of n/2 generatorsξα a generator denoted by ξ∗α with the following properties:

(ξα)∗ = ξ∗α , (ξ∗α)∗ = ξα . (2.8)

For λ ∈ C, we have (λξα)∗ = λ∗ξ∗α.

• For the conjugation of a product we require

(ξα1 · · · ξαn)∗ = ξ∗αnξ∗αn−1· · · ξ∗α1

. (2.9)

• Consider the case n = 2. The corresponding basis can be taken as 1, ξ, ξ∗, ξ∗ξ.

– If f is an analytic function (recall that an analytic complex function canbe expanded in a Taylor series and depends only on z but not on z∗),then

f(ξ) = f0 + f1ξ . (2.10)

– A general bilinear form is given by

A (ξ∗, ξ) = a0 + a1ξ + a1ξ∗ + a12ξ

∗ξ . (2.11)


– Derivatives. Here we require that

∂ξα∂ξβ

= δαβ . (2.12)

Then,

∂

∂ξ(ξ∗ξ) =

∂

∂ξ(−ξξ∗) = −ξ∗ ,

→ ∂

∂ξA (ξ∗, ξ) = a1 − a12ξ

∗ ,

∂

∂ξ∗∂

∂ξA (ξ∗, ξ) = −a12 = − ∂

∂ξ

∂

∂ξ∗A (ξ∗, ξ) ,

→

∂

∂ξ∗,∂

∂ξ

= 0 . (2.13)

– Integrals. The following rules are introduced:∫dξ = 0 ,

∫dξ∗ = 0 ,∫

dξ ξ = 1 ,

∫dξ∗ ξ∗ = 1 . (2.14)

Then, the integral of f(ξ) leads to∫dξ f(ξ) =

∫dξ (f0 + f1ξ) = f1 , (2.15)

and for A (ξ∗, ξ) we have∫dξ A (ξ∗, ξ) = a1 − a12ξ

∗ ,∫dξ∗A (ξ∗, ξ) = a1 + a12ξ ,∫

dξ∗ dξ A (ξ∗, ξ) = −a12 = −∫

dξ dξ∗A (ξ∗, ξ) . (2.16)

We see that with the adopted rules, differentiation and integration act inthe same way. The generalization of the rules above to n > 2 and evenis straightforward.

• δ-function. It is defined as

δ(ξ, ξ′) ≡∫

dη e−η(ξ−ξ′) = −(ξ − ξ′) , (2.17)

where η is also a Grassmann variable. We can easily verify that it works as aδ-function:∫

dξ′ δ(ξ, ξ′)f(ξ′) = −∫

dξ′ (ξ − ξ′)(f0 + f1ξ′) = f(ξ) . (2.18)


• Scalar product. We can define here the scalar product in a way that on theone hand resembles the form of a scalar product with bosonic coherent states,and on the other hand, allows functions of Grassmann variable to have thestructure of a Hilbert space.

(f, g) ≡∫

dξ∗ dξ e−ξ∗ξf ∗(ξ∗) g(ξ∗)

=

∫dξ∗ dξ (1− ξ∗ξ)(f ∗0 + f ∗1 ξ)(g0 + g1ξ

∗)

= f ∗0 g0 + f ∗1 g1 . (2.19)

2.2 Coherent states for fermions

After introducing Grassmann variables, we still need to deal with the commutationrelations between Grassmann variables and fermionic operators. For that purposewe associate ξα to the annihilation operator fα and ξ∗α to f †α, and require

ξα, fβ = ξ∗α, fβ = ξα, f †β = ξ∗α, f†β = 0 , (2.20)

and

(ξαfβ)† = f †β ξ∗α , etc . (2.21)

Since we already know how bosonic coherent states are, we try with a similarform for fermionic coherent states:

|ξ〉 = e−Pα ξαf

†α |0〉 =

∏α

(1− ξαf †α)|0〉 . (2.22)

Applying fα on this state we have

fα|ξ〉 = fα∏β

(1− ξβf

†β

)|0〉

=∏β 6=α

(1− ξβf

†β

)fα(1− ξαf †α

)|0〉︸︷︷︸

=ξα|0〉=ξα(1−ξαf†α)|0〉

= ξα|ξ〉 . (2.23)

Hence |ξ〉 is an eigenstate of fα, i.e. it is a coherent state. Once we have a coherentstate, we can see that its adjoint is a left eigenstate of the creation operator:

〈ξ|f †α = 〈ξ|ξ∗α . (2.24)

As in the bosoniuc case we can consider the application of a creation operator on acoherent state.

f †α|ξ〉 = f †α∏β 6=α

(1− ξβf

†β

)|0〉

= − ∂

∂ξα

(1− ξαf †α

)∏β 6=α

(1− ξβf

†β

)|0〉 = − ∂

∂ξα|ξ〉 , (2.25)


and in the same way we have

〈ξ|fα =∂

∂ξ∗α〈ξ| . (2.26)

The overlap of two coherent states is

〈ξ|ξ′〉 = 〈0|∏α

(1 + ξ∗αfα)(1− ξ′αf †α

)|0〉

= 〈0|∏α

(1 + ξ∗αξ′α) |0〉 = e

Pα ξ∗αξ′α . (2.27)

After having seen the properties of the fermionic coherent states, we considerthe resolution of unity. For that we just look at an expression similar to the one forbosons.

A =

∫ ∏α

dξ∗αdξα exp

(−∑α

ξ∗αξα

)|ξ〉〈ξ| . (2.28)

Next let us consider the overlap between two elements in the basis of Fock-space,

|α1, . . . , αn〉 = f †α1· · · f †αn|0〉 ,

|β1, . . . , βm〉 = f †β1· · · f †βm |0〉 . (2.29)

Their ovelap is

〈α1, . . . , αn|β1, . . . , βm〉 = δnm P

(α1, . . . , αnαi1 , . . . , αin

)δαi1β1 · · · δαinβn , (2.30)

where P(α1,...,αnαi1 ,...,αin

)= (−1)p with p the parity of the permutation relating α1, . . . , αn

and αi1 , . . . , αin . Now we can look at the matrix element of A between those twostates. Here we have on the one hand

〈α1, . . . , αn|ξ〉 = 〈0|fαn · · · fα1|ξ〉 = ξαn · · · ξα1 , (2.31)

since the number of permutations at the end is even. In the same way we have

〈ξ|β1, . . . , βm〉 = ξ∗β1· · · ξ∗βn , (2.32)

such that

〈α1, . . . , αn|A|β1, . . . , βm〉 =

∫ ∏α

dξ∗αdξα∏α

(1− ξ∗αξα)

×ξαn · · · ξα1ξ∗β1· · · ξ∗βn . (2.33)

For each index α, the possible combinations that can appear are

∫dξ∗αdξα (1− ξ∗αξα)

ξαξ∗α

ξ∗αξα1

=

1001

. (2.34)


Therefore, non-vanishing results are obtained only when some αi equals some βj,leading to m = n, and such that we can pair the αi’s and βj’s. However, since weare dealing with Grassmann variables, the order matters. This can be taken intoaccount as follows:

ξ∗β1· · · ξ∗βn = P

(α1, . . . , αnαi1 , . . . , αin

)δαi1β1 · · · δαinβnξ

∗α1· · · ξ∗αn . (2.35)

Inserting all this into (2.33), and comparing with (2.30), we have finally the resolu-tion of unity for fermionic coherent states,∫ ∏

α

dξ∗αdξα exp

(−∑α

ξ∗αξα

)|ξ〉〈ξ| = 1 . (2.36)

As a last remark before going over to the path-integral, we discuss here Gaussianintegrals, both for bosons and for fermions, since such operations are the ones mostcommonly carried out with path-integrals.

2.2.1 Gaussian integrals

For a single real variable, the Gaussian integral is∫ ∞−∞

dx e−ax2

=

√π

a. (2.37)

This result can be generalized for n variables multiplying a real symmetric positivedefinite matrix A. We consider the following integral:

I =

∫ ∞−∞

dx1 · · · dxn exp

(−1

2xiAijxj + xiJi

), (2.38)

where summation over repeated indices is understood. We first displace the argu-ment of the exponential by a change of variables

yi = xi − A−1ij Jj , (2.39)

such that

−1

2xiAijxj + xiJi = −1

2yiAijyj + JiA

−1ij Jj . (2.40)

We are left with the integral over yi’s. Since A is real and symmetric, it can bediagonalized by an orthonormal transformation, leading to∫ ∞

−∞dy1 · · · dyn exp

(−1

2yiAijyj

)=

n∏i=1

∫ ∞−∞

dzi e−ai

2z2i

=n∏i=1

√2π

ai=

(2π)n/2√detA

. (2.41)


Finally, we have

I =(2π)n/2√

detAexp

(JiA

−1ij Jj

). (2.42)

The calculations above can be easily generalized to complex (bosons) and Grass-mann fields (fermions).

• Gaussian integrals for bosons. In this case a hermitic positve matrix H for theintegral

Ib =

∫ n∏i=1

dφ∗i dφi2πi

exp(−φ∗iHijφj + J∗i φi + φ∗iJi

). (2.43)

As before, a change of variables can be performed and H can be diagonalized.For each eigenvalue we have then the integral∫

dz∗i dzi2πi

e−z∗i hizi =

∫dui dviπ

e−hi(u2i+v

2i ) =

1

hi, (2.44)

where we set zi = ui + ivi. Then, we have

Ib =1

detHexp

(J∗i H

−1ij Jj

). (2.45)

• Gaussian integrals for fermions. As in the case of bosons we consider a her-mitic positive matrix H. The integral is

If =

∫ n∏i=1

dη∗i dηi exp(−η∗iHijηj + ζ∗i ηi + η∗i ζi

), (2.46)

where η∗i , ηi, ζ∗i , and ζi are Grassmann variables. As before, a shift of variables

can be performed, and after a unitary transformation that diagonalizes H, wearrive at the following integral for each eigenvalue∫

dξ∗i dξi e−xi∗i hixii =

∫dξ∗i dξi (1− ξ∗i hiξi) = hi . (2.47)

Hence, the final result is

If = detH exp(ζ∗iH

−1ij ζj

). (2.48)

2.3 Path-integrals for fermions

Now we have all the ingredients to set up the path-integral for the Hamiltonian (2.1).As in the case of bosons, we consider the partition function in the grand-canonicalensemble.

Z = Tr e−βK , (2.49)


where the operator K is the equivalent of (1.42) but with the Hamiltonian (2.1).To take the trace, we consider the basis of occupation numbers in Fock-space, suchthat

(2.49) =∑|n〉

〈n|e−βK |n〉

=

∫ ∏α

dξ∗α dξα exp

(−∑α

ξ∗αξα

)∑|n〉

〈n|ξ〉〈ξ|e−βK |n〉 . (2.50)

Since we have states |n〉 with a definite occupation, they are given by

|n〉 = f †α1· · · f †αn|0〉 , (2.51)

and the matrix elements have the form

〈n|ξ〉 = ξαn · · · ξα1 ,

〈ξ|e−βK |n〉 =∑βi

cβ1,...,βnξ∗β1· · · ξ∗βn . (2.52)

Then,

〈n|ξ〉〈ξ|e−βK |n〉 → ξαn · · · ξα1ξ∗β1· · · ξ∗βn

= (−1)nξ∗β1· · · ξ∗βnξαn · · · ξα1 → 〈−ξ|e−βK |n〉〈n|ξ〉 , (2.53)

such that

Z =

∫ ∏α

dξ∗α dξα exp

(−∑α

ξ∗αξα

)∑|n〉

〈−ξ|e−βK |n〉〈n|ξ〉 . (2.54)

We see from the above, that in the case of fermions we have antiperiodic boundaryconditions in imaginary time, as oppossed to bosons. This feature resulting from theanticommutation relations of fermionic operators can be found in other formulations(see e.g. [6]).

Now we proceed as in the bosonic case, performing a Trotter slicing, normalordering, and inserting the resolution of unity between the time sclices. The resultis

Z = limL→∞

∫ L∏`=1

∏α

dξ∗α,` dξα,` exp

(−

L∑`=1

∑α

ξ∗α,`ξα,`

)

×L∏`=1

〈ξ`|1−∆τK +O(∆τ 2

)|ξ`−1〉 , (2.55)

where we used the convention that ξα,L = −ξα,0, and ξ∗αL = −ξα,0. The matrix

elements of K(f †α, fβ

)between coherent states are easily obtained, as in the bosonic

case.

〈ξ`|K(f †α, fβ

)|ξ`−1〉 = K

(ξ∗α, ξβ

)〈ξ`|ξ`−1〉 , (2.56)


where now K is a Grassmann valued function of ξ∗α, ξα. Recalling that the overlapbetween coherent states is given in (2.27), we have finally,

Z = limL→∞

∫ L∏`=1

∏α

dξ∗α,` dξα,` e−S(ξ∗,ξ) , (2.57)

where

S(ξ∗, ξ) =L∑`=2

∆τ

∑α

ξ∗α,`

[(ξα,` − ξα,`−1)

∆τ− µξα,`−1

]+H

(ξ∗α,`, ξα,`−1

)

+∆τ∑α

ξ∗α,1

[(ξα,1 + ξα,L)

∆τ+ µξα,L

]+H

(ξ∗α,1,−ξα,M

). (2.58)

At this point it is instructive to see the differences between the fermionic and thebosonic case. As we mentioned already for bosons, the discrete form is the one thatshould be used for actual calculations. Again, the usual notation, that makes aconnection with a field theory, is assuming continuity of the fields, resulting in

Z =

∫Dξ∗α(τ)Dξα(τ) e−S , (2.59)

with ξα(β) = −ξα(0), and the action given by

S =

∫ β

0

dτ∑α

[ξ∗α(τ)

(∂

∂τ− µ

)ξα(τ) +H (ξ∗α(τ), ξα(τ))

]. (2.60)

In order to obtain correlation functions, we can introduce source terms, as inthe bosonic case. As already seen in Sec. 2.2.1, the source terms coupling to linearforms of Grassmann variables have to be themselves Grassmanian. The action hasthe form

S → S +

∫dτ ddx

∑α

[J∗α(x, τ)ξα(x, τ) + ξ∗α(x, τ)Jα(x, τ)] , (2.61)

where now J∗α and Jα are Grassmann-valued functions. The derivatives of the log-arithm of the generating functional have then to be taken following the rules forderivation with Grassmann variables.

Problem: Bose-Einstein and Fermi-Dirac statistics

Consider a Hamiltonian for free particles

H =∑α

εαc†αcα , (2.62)

where c†α and cα are creation and annihilation oeprators, repectively, for eitherbosons or fermions. Calculate the partition function using coherent states for


a) bosons,

b) fermions,

using the discrete form of the action (1.48) for bosons and (2.58) for fermions.

Hint: write the action as a bilinear form, and write explicitly the correspondingmatrix for each case. For an L× L matrix, the determinant should be

detM (α) = 1 + (−1)L(s)L , bosons ,

detM (α) = 1− (−1)L(s)L , fermions , (2.63)

where s are the elements different from zero and one.From the form obtained for the partition functions calculate the respective oc-

cupation numbers.

Chapter 3

Path integrals for spins

After having constructed path-integrals for bosons and fermions, it could seem thatwe should be able to set up the path integral for arbitrary systems, since bosonsand fermions are the elementary degrees of freedom that constitute matter (as weknow it). However, there are situations, where further degrees of freedom play thecentral role. This is the case for magnetic systems, where only the spin degreesof freedom are of interest. In constructing coherent states, we will have a more”classical” picture at hand, although we are going to be dealing with quantum spinsystems. A more expanded exposition about coherent states related to different Liegroups can be found in the book by A. Perelomov [7].

3.1 Coherent states for spins

Let us recall some facts about spin operators in order to set the notation for later.Spin operators fulfill the SU(2) algebra

[Sα, Sβ] = i~εαβγSγ , α, β, γ = 1, 2, 3 (x, y, z) . (3.1)

Furthermore, S2 =∑

α S2α commutes with Sα ∀ α, such that we consider the set of

states that are eigenstates of S2 and S3:

S2|S,m〉 = ~2S(S + 1)|S,m〉 ,S3|S,m〉 = ~m|S,m〉 , (3.2)

where −S ≤ m ≤ S and S = 1/2, 1, 3/2, . . . . Further we can introduce operatorsS± = S1 ± iS2, such that

[S3, S±] = ±~S± . (3.3)

Each of the values of S corresponds to an irreducible representation, where the spinoperators are represented by matrices of dimension (2S+1)×(2S+1). The completeset of states for each S can be obtained by taking the so-called lowest (highest) state

27


|S,−S〉 (|S, S〉) and operating on them with S+ (S−), where

S+|S,m〉 =√

(S −m)(S +m+ 1) |S,m+ 1〉 ,S−|S,m〉 =

√(S +m)(S −m+ 1) |S,m− 1〉 . (3.4)

Having the complete set of states, we can construct the corresponding matrices bytaking matrix elements of Sα in those states.

After setting up the notation, let us discuss some properties of the group SU(2)that gives us a geometrical perspective, that will be useful later, when discussingcoherent states and also systems of many spins. We start by recalling that SU(2) isthe group of 2× 2 unitary matrices with determinant 1:

g =

(α β

−β∗ α∗

), with |α|2 + |β|2 = 1 . (3.5)

With α = α1+iα2 and β = β1+iβ2, the condition above means α21+α2

2+β21 +β2

2 = 1,such that SU(2) is homeomorphic to S3, the surface of a sphere in 4 dimensions.

Homeomorphism: given T1 and T2 continuous manifolds, α : T1 → T2 is continuousand its inverse is also continuous.

Putting β = 0, we see that we have a set that is also a group, that we call H =h

,where

h =

(α 00 α∗

), with |α|2 = 1 ⇒ α = eiψ . (3.6)

Hence, H ⊂ SU(2), is the subgroup H = U(1).We can now go back to our original interest, namely to generate coherent states

for spins. We have seen that in the case of bosons and fermions, we generated thecorresponding coherent states from the vacuum. Thinking in terms of the operatorsS±, we can take the lowest (highest) weight state as the vacuum. For definitness wetake |S,−S〉 as the state from which we expect to generate the whole Hilbert space.For the moment, let us restrict ourselves to the case S = 1/2, and see what happensif some h operates on the lowest weight state:

h

∣∣∣∣12 ,−1

2

⟩= e−iψ

∣∣∣∣12 ,−1

2

⟩, (3.7)

i.e. such an operation brings a phase, and we are not generating a new state. Inorder to avoid such a trivial phase, we consider the left coset of H.

Left coset of H: For an element g ∈ G and a subgroup H ⊂ G, gH = gh, h ∈ H.


We can eliminate the redundancy that the elements of H bring in, by considering allgh ∈ gH as equivalent. Such an equivalence relation defines a quotient space (≡ theset of left cosets), that in our case is X =SU(2)/U(1). In order to see its structure,we consider explicitely its elements:(

α β−β∗ α∗

)(eiψ 00 e−iψ

)=

(ραei(φα+ψ) ρβei(φβ−ψ)

−ρβe−i(φβ−ψ) ραe−i(φα+ψ)

), (3.8)

where we can take always ψ = −φα. We can generate another set of elements disjointfrom the first one by taking ψ = −φα + π. Hence, given the equivalence relation,the quotient space is given by the set

X =

(α β

−β∗ α

), α ∈ R , α2 + β2

1 + β22 = 1 , (3.9)

that is, X ' S2. A possible parametrization of X is

α = cosθ

2, β = − sin

θ

2e−iϕ , 0 ≤ θ ≤ π , 0 ≤ ϕ < 2π . (3.10)

On the other hand, each point on S2 corresponds to the unit vector

n = (cosϕ sin θ, sinϕ sin θ, cos θ) . (3.11)

Therefore, we can view each point in X as obtained from the north pole by rotatingn by an angle θ around an axis perpendicular to the plane formed by the line fromthe center of the sphere to the north pole and the vector n. Given the vector n,the axis on the equatorial plane with the properties described above is given bym1 = sinϕ and m2 = − cosϕ. The corresponding SU(2) rotation is given by

gn = exp[iθ(m1

σ1

2+m2

σ2

2

)], (3.12)

where σi, i = 1, 2, 3, are the Pauli Matrices. Indeed,

exp[iθ(m1

σ1

2+m2

σ2

2

)]=

∑n

1

n!

(iθ

2

)n(m1σ1 +m2σ2)n

=∑n even

1

n!

(iθ

2

)n+ (m1σ1 +m2σ2)

∑nodd

1

n!

(iθ

2

)n=

(cos θ

2− sin θ

2e−iϕ

sin θ2eiϕ cos θ

2

). (3.13)

Next we generalize the discussion above, that was performed using S = 1/2, toarbitrary values of S. Since the geometric picture we used did not depend on S, we


preserve its essence in going to general S. Then, we examine states created fromthe lowest weight state in an analogous form as before:

|n〉 = exp (i θm · S) |S,−S〉 , (3.14)

where we recall that m = (sinϕ,− cosϕ, 0), such that

m · S = m1S1 +m2S2 =i

2

(e−iϕS+ − eiϕS−

). (3.15)

Defining ξ = − θ2e−iϕ, we can write

exp (i θm · S) = exp (ξS+ − ξ∗S−) . (3.16)

Since S± act as creation and annihilation operators, it is useful to go over to a formreminiscent of normal ordering. Here we have

exp (ξS+ − ξ∗S−) = eζS+eηS3eζ′S− , (3.17)

where we defined

ζ = − tanθ

2e−iϕ ,

ζ ′ = −ζ∗ ,

η = −2 ln cosθ

2. (3.18)

We can easily verify (3.17) in the case S = 1/2.

eζS+eηS3eζ′S− =

∞∑n=0

ζn

n!Sn+

∞∑m=0

ηm

m!Sm3

∞∑p=0

ζ ′p

p!Sp−

= (1 + ζS+)(

coshη

2+ σ3 sinh

η

2

)(1 + ζ ′S−)

=

(eη/2 + e−η/2ζ ′ζ ζe−η/2

ζ ′e−η/2 e−η/2

)= (3.13) . (3.19)

Now we can consider the states generated by (3.17) out of the lowest weight state,in the general case.

exp (i θm · S) |S,−S〉= eζS+eηS3eζ

′S−|S,−S〉

= eζS+

∞∑n=0

ηn

n!Sn3 |S,−S〉︸︷︷︸=(−S)n|S,−S〉

=

(cos

θ

2

)2S ∞∑n=0

ζn

n!Sn+|S,−S〉

=

(cos

θ

2

)2S 2S∑n=0

ζn[

(2S)!

(2S − n)!n!

]1/2

|S,−S + n〉 = |θ, ϕ〉 . (3.20)


It remains to see if the states form an overcomplete set, i.e. the resolution of unity.Since the states created are parametrized by the coordinates of the unit sphere, wehave to integrate over the measure of spherical coordinates. This leads to∫

sin θdθ dϕ |θ, ϕ〉〈θ, ϕ|

=2S∑

m,n=0

[(2S)!

(2S −m)!m!

]1/2 [(2S)!

(2S − n)!n!

]1/2

×∫ π

0

sin θdθ

(cos

θ

2

)4S (− tan

θ

2

)m(− tan

θ

2

)n×∫ 2π

0

dϕ e−imϕeinϕ|S,−S +m〉〈S,−S + n|

= 2π2S∑n=0

(2S)!

(2S − n)!n!

∫ π

0

sin θdθ

(cos2 θ

2

)2S(

sin2 θ2

cos2 θ2

)n

×|S,−S + n〉〈S,−S + n|

= 2π(2S)!2S∑n=0

1

(2S − n)!n!

∫ π

0

sin θdθ

(1 + cos θ

2

)2S (1− cos θ

1 + cos θ

)n×|S,−S + n〉〈S,−S + n|

=2π(2S)!

22S

2S∑n=0

1

(2S − n)!n!

∫ 1

−1

du (1 + u)2S−n (1− u)n︸︷︷︸≡I

×|S,−S + n〉〈S,−S + n| , (3.21)

where

I =22S+1

2S + 1

(2S − n)!n!

(2S)!, (3.22)

such that

(3.21) =4π

2S + 1

2S∑n=0

|S,−S + n〉〈S,−S + n| . (3.23)

Then, the resolution of unity is given by

1 =2S + 1

4π

∫sin θdθ dϕ |θ, ϕ〉〈θ, ϕ| . (3.24)

3.2 Path-integrals for quantum spin systems

Once we arrived at an overcomplete set of spin-states, we can go over to a partitionfunction. In the following we use as a short notation |n〉 = |θ, ϕ〉. The partition


function is given by

Z = Tre−βH =

∫dn 〈n|e−βH |n〉 , (3.25)

where

dn =2S + 1

4πsin θdθdϕ . (3.26)

Now we can just repeat the same steps we had in the cases of bosons and fermions.We start by a Trotter-slicing:

e−βH =L∏`=1

e−∆τH , ∆τ =β

L, (3.27)

such that the partition function goes over into

Z = limL←∞

∫ L∏`=1

dn`

L∏`=1

〈n`|e−∆τH |n`−1〉 , (3.28)

where n0 = n and nL = n, i.e. we have periodic boundary conditions in imaginarytime. For the matrix elements we have up to O (∆τ 2),

〈n`|e−∆τH |n`−1〉 ' 〈n`|1−∆τH|n`−1〉

= 〈n`|n`−1〉[1−∆τ

〈n`|H|n`−1〉〈n`|n`−1〉

]' 〈n`|n`−1〉 exp

[−∆τ

〈n`|H|n`−1〉〈n`|n`−1〉

]' 〈n`|n`−1〉 exp

[−∆τ

〈n`|H|n`〉〈n`|n`〉

]. (3.29)

The factor in the denominator of the exponent could be a problem, but it can beeasily seen that the states |n〉 are properly normalized:

〈n`|n`〉 =

(cos

θ

2

)4S 2S∑n=0

(ζ∗ζ)n(2S)!

(2S − n)!n!

=

(cos

θ

2

)4S

(1 + ζ∗ζ)2S = 1 . (3.30)

We can use the fact above to rewrite the overlap between coehrent states on differenttime slices in a more convenient form:

L∏`=1

〈n`|n`−1〉 =L∏`=1

[1− 〈n`| (|n`〉 − |n`−1〉)]

' exp

[−

L∑`=1

∆τ〈n`|∂

∂τ|n`〉

]. (3.31)


Finally, the partition function can be written as

Z =

∫Dn e−S , (3.32)

where

S =

∫ β

0

dτ

[〈n| ∂

∂τ|n〉+ 〈n|H|n〉

]. (3.33)

Once we arrived at the partition function, we have still to examine the matrixelements appearing in the action. For a Hamiltonian containing only spin-operators,we need only to look at their expectation values in terms of coherent states.

i) 〈n|S3|n〉.

〈n|S3|n〉 =

(cos2 θ

2

)2S 2S∑n=0

(ζ∗ζ)n(2S)!

(2S − n)!n!〈S,−S + n| S3|S,−S + n〉︸︷︷︸

=(−S+n)|S,−S+n〉

= −S +

(cos2 θ

2

)2S 2S∑n=0

(ζ∗ζ)n(2S)!n

(2S − n)!n!

= −S +

(cos2 θ

2

)2S

2S ζ∗ζ︸︷︷︸sin2 θ

2/ cos2 θ

2

2S−1∑n=0

(ζ∗ζ)n(2S − 1)!n

(2S − 1− n)!n!︸︷︷︸(1+ζ∗ζ)2S+−1

= −S + 2S sin2 θ

2= −S cos θ . (3.34)

ii) 〈n|S+|n〉 = −S sin θ eiϕ.

iii) 〈n|S−|n〉 = −S sin θ e−iϕ.

Hence we have in short

〈n|S|n〉 = −Sn , (3.35)

an expression that gives a classical picture for the spin. For the Heisenberg Hamil-tonian, for instance, we have

〈n|H|n〉 = JH∑〈i,j〉

〈n|Si · Sj|n〉

= JHS2∑〈i,j〉

ni · nj , (3.36)

and at least the Hamiltonian part is indistinguishable from a classical Hamiltonian.In fact, without the first term in the integrand of the action (3.33), the partitionfunction (3.32) is the one corresponding to a d + 1 dimensional classical system,where the extra dimension comes from the imaginary time. This tells us, that thefirst term in the integrand of the action (3.33) contains the information about thequantum mechanical character of the system.


3.2.1 Berry phase

Due to the special role of the first term in the integrand of the action (3.33), weexamine it in a section of its own. We will see that it is intimately related toa concept introduced by Michael Berry some years ago [8], when considering thephase factor that arises when a Hamiltonian H(R) is transported around a circiutC in the space of the parameters R.

A direct calculation (not carried out here explicitely for brevity) shows that

〈n| ∂∂τ|n〉 = −iS(1− cos θ)ϕ , (3.37)

such that the partition function (3.32) acquires a phase

exp

(−∫ β

0

dτ〈n| ∂∂τ|n〉)

= exp

(iS

∫ β

0

dτ (1− cos θ)ϕ

). (3.38)

Moreover, ∫ β

0

dτ∂ϕ

∂τ(1− cos θ) =

∫Γ

(1− cos θ) dϕ , (3.39)

where Γ is the closed orbit (due to periodic boundary conditions in imaginary time)described by n. The last expression makes evident that the phase does not dependon the speed to traverse the path Γ. Instead, the Berry phase is of purely geometricalnature.

Γ

Figure 3.1: Sketch of the path of n along Γ.


Since we are considering an integral over a closed path, it is tempting to writethis expression as the line integral of a vector field, that is∫

Γ

(1− cos θ) dϕ =

∫Γ

A · d` . (3.40)

In fact, taking spherical coordinates on a sphere of radius 1, we can write

xdy − ydx

z + 1=

sin2 θ

cos θ + 1dϕ = (1− cos θ)dϕ . (3.41)

Hence, the vector field is

A =1

r(z + r)(−y, x, 0) , (3.42)

where we generalized the expression to an arbitrary radius r. This generalizationwill allow us to understand the geometric meaning of the integral we are dealingwith.

Having the circulation of a vector field on a curve in three dimensions, we canapply Stokes’ theorem. For that purpose, we need the curl of the vector field, thatin our case is

∇×A =x

r3. (3.43)

If we think of A as a gauge field, then, the corresponding magnetic field is a radialone, and hence, the field of a magnetic monopole. Summarizing the discussion above,the Berry phase in the case of a spin is given by∫

Γ

A · d` =

∫F

x

r3· ds , (3.44)

where F is the surface with boundary Γ. The expression above corresponds to thesolid angle subtended by the surface ds, as depicted below. It is the projection onthe sphere of unit radius of the surface F :∫

F

x

r3· ds = Ω , (3.45)

such that ∫ β

0

dτ〈n| ∂∂τ|n〉 = −iSΩ . (3.46)

This last expression will put into evidence the quantum nature of the system, sincethe solid angle for a closed curve on S2 has two possible values, depending on theconvention we use for the direction of Γ. The two possibilities are

exp

(iS

∫ β

0

dτ (1− cos θ)ϕ

)= eiSΩ = e−iSΩ′ , (3.47)


x

dsθ

O

Figure 3.2: Solid angle seen from O subtended by the element of surface ds.

where Ω is the solid angle for one circulation direction, and Ω′ for the other one,with the due change of sign. But this means that

eiS(Ω+Ω′) = 1 . (3.48)

But being on a sphere, Ω + Ω′ = 4π, and the condition above can only be fulfilledif S = 0, 1/2, 1, 3/2, . . . . Hence, the Berry phase is responsible for the quantizationof S, making the difference with respect to a system with classical spins. Thequantization resulting from a magnetic monopole was addressed first by Dirac inorder to quantize the electric charge. A short and interesting discussion is given byR. Jackiw on occassion of the Dirac Memorial in 2002 [9].

Since the Hamiltonian part could be written entirely in terms of the n-fields, itwould be also desirable to do so in the case of the Berry phase. In this case we have∫ β

0

dτ〈n| ∂∂τ|n〉 = −iS

∫ β

0

A · ∂n∂τ

dτ , (3.49)

where the gauge field has to obey the condition

εabc∂

∂nbAc = na , (3.50)

with |n|2 = 1.

Problem: Quantum Heisenberg antiferromagnet in one dimension

Consider the Hamiltonian for a quantum Heisenberg antiferromagnet in one di-mension,

HH = J∑i

Si · Si+1 , (3.51)


where J > 0, and the spin operators may correspond to S = 12, 1, 3

2, . . . .

i) Write the corresponding partition function using coherent states.

ii) For β → ∞ we expect that the spins are close to a Neel state, with ni =(−1)in, n being a unit vector in some direction. Introduce a local orderparameter Ωi ∼ ni+1 − ni, that shows small variations on neighboring sites,and fluctuations perpendicular to it ì ∼ ni+1 +ni. In term of these fields weexpress the original spins as

ni = (−1)iΩi

√1− a2`2

i + aì , (3.52)

where a is the lattice constant and the factor multiplying Ωi enforces thatΩ2i = 1.

a) Introduce the proposed form for the spin-fields in the interaction part ofthe action and assume continuity such that

Ωi+1 = Ωi + a∂xΩi +1

2a2∂2

xΩi +O(a3),

ì+1 = ì + a∂xì +O(a2), (3.53)

and the summations over lattice sites go over to integrals. Since Ω2i = 1,

Ωi · ∂xΩi = 0, and Ωi · ∂2xΩi = −∂xΩi · ∂xΩi.

b) Introduce the proposed form for the spin-fields in the Berry phase andassume continuity as well. Notice that due to (3.50) A [Ω] = A [−Ω]. itis useful to notice that (3.50) implies

∂Ab

∂Ωa− ∂Aa

∂Ωb= εabc Ωc . (3.54)

At the end the action should read

SH = iS

4

∫dτ dx εµνεabc Ωa∂µΩb∂νΩ

c + iS

∫dτ dx ` · (Ω× ∂τΩ)

+JaS2

2

∫dτ dx

[∂xΩ · ∂xΩ + 4`2

]. (3.55)

iii) Since ` appears in a bilinear form, it can be integrated out, or what is equiva-lent, one can take the saddle-point with respect to `. Replacing the solution ofthe saddle-point equation in the action, a field-theory for Ω is attained. Bringit into the form

S =1

2g

∫dτ dx

[1

c(∂τΩ)2 + c (∂xΩ)2

]+iS

4

∫dτ dx εµνεabc Ωa∂µΩb∂νΩ

c . (3.56)

Which is the form of g and c in terms of the original parameters S, J , and a?

Chapter 4

Topological excitations inquantum spin systems

We have seen that the Hamiltonian part of a spin system could be written in terms ofclassical fields. In general, since quantum spin systems motivated from a condensedmatter point of view live on a lattice, we have to deal with fields that are not con-tinuous. This precludes in general an analytic treatment of such systems, calling fornumerical methods to deal with them if accurate and controlled results are required.However, in many cases, the interest is centered on possible phase transitions, whereone may suspect or knows that they are continuous ones. In this case, close to thetransition, the spin-spin correlation length diverges. This sign of criticality may helpto characterize the system further on an analytical basis, since we expect that thespins in close vecinity are strongly correlated, such that their configuration changesonly weakly with distance, i.e. the fields are expected to be well described by con-tinuous ones. A consequence of continuity can also be that topology starts to play arole, by protecting configurations that cannot be deformed continuously into trivialones. In this chapter we will discuss some of those configurations.

4.1 The non-linear σ-model

Let us start with the Heisenberg model we shortly mentioned in the previous chapter,

H = −J∑〈i,j〉

Si · Sj , (4.1)

that for simplicity we take as a ferromagnetic one (J > 0). For the moment werestrict ourselves to a classical model, and we write similarly to the results obtainedwith coherent states in the previous chapter, Si = Sni, with |ni|2 = 1. The partitionfunction can be written as

Z =∑Si

e−βH =∑ni

exp

g∑〈i,j〉

ni · nj

, (4.2)

39


where we defined g = βJS2. At high temperatures (g → 0), all configurations areequally important, so that the system is disordered. At low temperatures (g →∞),the favoured configuration is the one with ni · nj = 1, i.e. the ferromagnetic one,that breaks the rotational symmetry of the Hamiltonian.

Up to now, we did not specify the number of components of Si. If the classicalspin we are considering stems from a quantum mechanical one, we would haven = 3 components and the Hamiltonian (4.1) would have a global O(3) symmetry.However we could generalize the model to n any natural number. We restrict ushere to n > 1, since for n = 1 we have a global discrete symmetry (Z2), instead of acontinuous one. We have also to specify the dimension d of the lattice in (4.1). Sincewe are considering systems with continuous global symmetries (O(n)), we recall theMermin-Wagner theorem [10] that states that a continuous symmetry cannot bebroken spontaneously at finite temperature at d = 2. Hence, we consider systemswith d ≥ 2. For simplicity, we assume a hypercubic lattice.

Then, we expect that there is a coupling gc (finite in d ≥ 3 or zero in d = 2),where, on approaching it, the correlation length ξ diverges, or that a/ξ → 0, wherea is the lattice constant. Within the correlation length the spins are almost ordered,so that neighboring spins differ only by a small angle. This is equivalent to considerthe limit a → 0, and assume continuity of the spin-fields, such that a gradientexpansion is possible:

nj ' ni + ∂µniaµ +

1

2∂2µnia

µ2 + . . . , (4.3)

where µ = x, y, z, . . . the component corresponding to the link between i and j.Introducing this expression into the interaction we have

ni · nj ' 1 + ni · ∂µni︸︷︷︸=0

aµ +1

2ni · ∂2

µniaµ2 + . . . , (4.4)

such that∑〈i,j〉

ni · nj '1

2

∑i

a2ni · ∂2µni

' 1

2ad−2

∫ddxn · ∂2

µn = − 1

2ad−2

∫ddx ∂µn · ∂µn , (4.5)

leading to a partition function

Z =

∫Dn (x) e−S , (4.6)

with an action

S =1

2g

∫ddx ∂µn · ∂µn , (4.7)


where we defined the coupling constant

g =ad−2

g. (4.8)

This definition shows explicitely that for d = 2 g is dimensionless. In fact, at d = 2the model is scale invariant. The action above is the O(n) non-linear σ-model.

Although the action looks like the one of a free field, it is non-trivial due to theconstraint |n|2 = 1. Close to gc we can imagine that the spins are aligned along somedirection, whose component we call σ. The other components, describing transversefluctuations around σ are called π, such that

n = (π, σ) ⇒ σ =√

1− π2 . (4.9)

performing an expansion of the square root

σ = 1− 1

2π2 − 1

8

(π2)2 − 1

16

(π2)4 − · · · , (4.10)

and rescaling the fields π/√g → π, the integrand of the action becomes

1

2g∂µn · ∂µn → 1

2∂µπ · ∂µπ +

g

8∂µπ

2 · ∂µπ2 +g2

16∂µπ

2 · ∂µ(π2)2

+ · · · ,(4.11)

such that the original rotational symmetry is realized here in a non-linear way. Hencethe name of the model. Moreover, in the weak coupling limit, g → 0, the actionreduces to that of free fields giving quadratic fluctuations around the mean-fieldsolution, σ = 1.

4.2 Topological excitations

We will focus here on specific cases with respect to the number of components nand the dimensionality d, that will lead to topological excitations like vortices andskyrmions.

4.2.1 The XY-model and vortices

We discuss first the case n = 2, such that the Hamiltonian (4.1) reduces to theXY-model. Furthermore, we restrict ourselves to d = 2. In this case, due to theMermin-Wagner theorem we already mentioned, we should not expect spontaneoussymmetry breaking for g > 0. However, as shown by Kosterlitz and Thouless [11], atransition takes place at finite temperature, where in the low temperature phase, thespin-spin correlation function decays with a power-law, and hence, the correlationlength is infinite, but the order parameter is zero. The high temperature phase hasa finite correlation length. Such a transition is due to the presence of vortices and


antivortices that bind in the low temperature phase and unbind at the transition.We are not going to consider the transition directly (see e.g. [12]).

Let us assume that the temperature is low enough, such that the spins are almostordered or more precisely, such that ξ a. In that case we can use the action (4.7).For n = 2 we can easily solve the constraint of the field n, namely n = (cos θ, sin θ).But then, the action becomes

S =1

2g

∫ddx (∂µθ)

2 . (4.12)

In the partition function we need to have a finite action, such that the configurationis not exponentially suppressed. This means that for g small enough, θ should be aconstant at infinity. First we can consider the ’classical’ equation of motion, i.e. theone that corresponds to δS = 0 (since we started with a classical system, the termclassical now refers to a mean-field solution, i.e. disregarding fluctuations). Theresult is

4θ = 0 , (4.13)

that is the Laplace’s equation. The solutions are harmonic functions. Since we arein d = 2, we can identify space with the complex plane, i.e. θ(x, y)→ θ(z). Then, θcan be seen as the real part of an analytic function, since in this case the real andimaginary parts satisfy separately Laplace’s equation. However, if such a functionis an entire function (analytic everywhere), due to the Cauchy-Liouville theorem, itshould be a constant, that is the real part corresponds to a plain ferromagnet. Butthere is the possibility of having isolated singularities. Since θ is a phase, in fact thephase of nx + iny, a possibility is

eiθ =z − z0

|z − z0|→ θ = arctan

[Im (z − z0)

Re (z − z0)

]. (4.14)

One can directly corroborate that 4θ = 0 holds in this case. For simplicity we take

r

a

b

αϕ

Figure 4.1: Coordinates (x, y) around the singularity at (a, 0).

z0 = (a, 0), such that

Im (z − z0) = y = b sinα , Re (z − z0) = x− a = b cosα , (4.15)


where b and α are displayed in Fig. 4.1. Then, we have θ = α + 2nπ. This resultshows on the one hand, that θ is not single-valued. This is not a problem, since thephysical variable is the spin, that is single-valued. On the other hand, the result doesnot obey the boundary conditions we have chosen, i.e. θ = θ∞ at infinity. We will tryto remedy this problem later. Nevertheless, we can learn from the obtained result,the physical meaning of the singularity, by looking at the spin field at some distanceb from it. The spin configurations are given by n = (cosα, sinα), that is they pointoutwards from the circle with radius b. Since a global rotation, say by π/2, is alsoa possible solution, we can see now that the introduction of a singularity leads toa vortex in the spin field. Once we understood the meaning of the singularity, weintroduce a configuration that respects the boundary conditions at infinity. Now wepropose for θ the following form:

eiθ = eiθ∞(z − z+)

|z − z+||z − z−|(z − z−)

, (4.16)

where now we have two singularities located at z±. In the limit |z| → ∞ the phasescancel each other, and we have θ → θ∞. In order to obtained the correspondingspin-configuration let us choose z± = (±a, 0). Then, we have

θ = θ∞ + Arg

[(z − z+)

|z − z+||z − z−|(z − z−)

]= θ∞ + arctan

(2aImz

|z|2 − a2

). (4.17)

In order to see now the spin configuration, we consider a circle of radius b aroundeach singularity, and a point on the circles corresponds to angles α±. Then on eachcircle we have

Imz = b sinα± ,

|z|2 − a2 = b(b± 2a cosα±

), (4.18)

such that on each circle

θ± = θ∞ + arctan

(2a sinα±

b± 2a cosα±

). (4.19)

Assuming b 2a, when α± go from 0 to 2π, θ+ increases by 2π, while θ− decreasesby 2π. Going to the spin configurations as before we can visualize the spin-fields asoutgoing arrows on one circle and some ingoing ones on the other. By performingagain a global rotation by π/2, we arrive at the more familiar picture for a vortexon one circle and an antivortex on the other one.

As pointed out by Kosterlitz and Thouless [11], the necessity of a vortex antivor-tex pair can also be explained energetically, since a single vortex would lead to alogarithmically divergent energy, a divergence that is cut off when an antivortex isincluded.


After seeing how vortices appear, let us put them into a general frame concern-ing the topological properties. For this, we introduce first in a loose way someterminology. Topology is about describing properties of spaces that can be seen asequivalent if they can be brought from one to another in a continuous way. Thiscan be formalized with the notion of a homotopy.

Homotopy: given f and g continuous functions from a (topological) spaceX to a(topological) space Y , the continuous function H : X × [0, 1]→ Y ,such that for x ∈ X H(x, 0) = f(x) and H(x, 1) = g(x) is ahomotopy.

With that notion we can define homotopy sectors, where each sector is given by ahomotopy, i.e. all the mappings that can be continuously deformed into one anotherare identified in an equivalence class. On the basis of such equivalence classes,homotopy groups are defined, that take all the mappings from a sphere Sn onto aspace X, and denoted by πn(X). In the case of the vortices we encountered above,we have π1(S1) = Z, since the mapping from one circle to the other can be performedby covering it an integer number of times. For a given configuration we can obtainthe corresponding homotopic sector by calculating the vorticity as follows:∮

∇θ · d` = 2πq , (4.20)

where the vorticity q ∈ Z counts the number of times θ winds around when we goonce around the vortex.

4.2.2 The O(3) non-linear σ-model and skyrmions

We consider now the case n = 3. Before we start to look for topological non-trivialfield configurations, let us summarized how we found them in the case n = 2. Therewe had a trivial solution, where n is a constant in space, i.e. the ferromagneticcase. However, we found non-trivial finite energy solutions of the classical equationsof motion with a topological character. If we consider fluctuations around thoseconfigurations, and assuming that they are small, they will amount to continuousdeformations of the classical solutions, that will not be able to change the homotopysector. Hence, it suffices to consider the classical equations of motion. We stay atd = 2 and consider the action (4.7). However, now with n = 3 it is not easy tosolve the constraint. Therefore, we introduce now a Lagrange multiplier to take theconstraint |n|2 = 1 into account. The action is now

S →∫

d2x

[1

2∂µn · ∂µn+ λ (n · n− 1)

], (4.21)


where we disposed of the coupling constant, since it will play no role in the discussion.The Euler-Lagrange equations are now

4n+ λn = 0 , (4.22)

that we can use to eliminate λ using the constraint on n:

λ = λn · n = −n · 4n , (4.23)

such that the equation of motion has the form

4n− (n · 4n)n = 0 . (4.24)

For a given solution, the energy (or action, since the action comes originally from aclassical statistical problem) is

E =1

2

∫d2x ∂µn · ∂µn . (4.25)

The trivial solution is the one with E = 0. As in the previous section, we look nowfor solutions of (4.24) with finite E. To find them, we recall a trick introduced byBelavin and Polyakov [13]. Starting from the identity∫

d2x [(∂µn± εµνn× ∂νn) · (∂µn± εµνn× ∂νn)] ≥ 0 . (4.26)

Carrying out the multiplication we obtain∫d2x ∂µn · ∂µn ≥ ±

∫d2x εµνn · (∂µn× ∂νn) , (4.27)

such that defining

Q =1

8π

∫d2x εµνn · (∂µn× ∂νn) , (4.28)

we have a lower bound for E depending on Q:

E ≥ 4π|Q| . (4.29)

Our next step is to discuss the meaning of Q.First we notice that the mixed product in (4.28) corresponds to a surface element

of the sphere, since n is a unit vector. In fact, for a directed surface element wehave

dsa = εabcdnb dnc . (4.30)

On the other hand, in order to have configurations with a finite action, we requireas in the case n = 2, that the fields at infinity point all in the same direction. This


means that R2 can be compactified to S2, the mapping between the two spaces beinga stereographic projection. Then, we arrive at a situation similar to the previous onefor n = 2, but instead of π1(S1), we should be considering π2(S2), i.e. the possiblecoverings of a sphere by another one. To show that this is so, we consider the changeof variable in going from cartesian to spherical coordinates for n . In doing so, thesurface element above can be written as

dsa =1

2ε`mεabc

∂nb∂ξ`

∂nc∂ξm

d2ξ , (4.31)

where we denote the two angular variables by (ξ1, ξ2). Then, Q that is given by anintegral over space, can be brought over to an integral over internal space (i.e. theangular variables):

Q =1

8π

∫d2x εµνεabc n

a∂µnb∂νn

c

=1

8π

∫d2x εµνεabc n

a∂nb

∂ξ`

∂ξ`∂xµ

∂nc

∂ξm

∂ξm∂xν

=1

8π

∫d2ξ ε`mεabc n

a∂nb

∂ξ`

∂nc

∂ξm, (4.32)

where we used that

d2x εµν∂ξ`∂xµ

∂ξm∂xν

= ε`md2ξ , (4.33)

is the Jacobian for the transformation from (x1, x2) to (ξ1, ξ2). Then, with (4.31)we have finally

Q =1

4π

∫nadsa =

1

4π

∫dΩ = n ∈ Z . (4.34)

The result above shows that Q counts the covering of the sphere that resulted fromcompactifying space by the sphere swept by n. The lower bound for E that wefound in (4.29) corresponds to the energy that is minimized by in each topologicalsector.

We still need to see how such field configurations look like. The configurationsthat correspond to the lower bound for E are those where the equality in eq. (4.26)is fulfilled. This leads to a differential equation for n:

∂µn = ∓εµνn× ∂νn . (4.35)

Since the solutions of this differential equation minimize the energy (or action),they are also solutions of (4.24). They do not represent the most general solution,but characterize the topological sectors. Since at some point we mentioned thestereographic projection as a mean to connect the points on a sphere to points inthe plane, we examine it closer.


Figure 4.2 illustrates the stereographic projection of n from the unit sphere tothe plane, where the points are denoted by (ω1, ω2). With the conventions adoptedwe have

tan γ =1− n3

n1

, (4.36)

such that

n 1

n 3

ω1

γ

x

z

Figure 4.2: Stereographic projection.

ω1 =2n1

1− n3

,

ω2 =2n2

1− n3

. (4.37)

It is convenient to pass to the complex plane with the complex coordinate ω =ω1 + iω2. Also the space on which n lives can be viewed as the complex plane withcoordinates z = x1 + ix2. Let us now go back to the differential equation (4.35) thatwe want to solve. For the two components entering ω in the numerator, we have

∂1n1 = ∓(n2∂2n3 − n3∂2n2) ,

∂1n2 = ∓(n3∂2n1 − n1∂2n3) ,

∂2n1 = ±(n2∂1n3 − n3∂1n2) ,

∂2n2 = ±(n3∂1n1 − n1∂1n3) , (4.38)


and since

∂µωi =2

(1− n3)2(∂µni + ni∂µn3 − n3∂µni) , i = 1, 2 , µ = 1, 2 , (4.39)

we have finally

∂1ω1 =2

(1− n3)2(∂1n1 ∓ ∂2n2) = ∓∂2ω2 ,

∂2ω1 =2

(1− n3)2(∂2n1 ± ∂1n2) = ±∂1ω2 . (4.40)

That is, the differential equation (4.35) has as solutions analytic functions ω(z),since the equations above are the Cauchy-Riemann conditions. The upper sign isfor ω(z∗) and the lower sign for ω(z). However, if we are dealing with an entirefunction that is bounded everywhere, then due to the Cauchy-Liouville theorem,the function is a constant. Actually, we have chosen as a boundary condition n3 = 1at infinity, so that at infinity ω →∞, therefore ω is not bounded everywhere. Sincean analytic function can be expanded in a Taylor series, we can consider monomialsas possible solutions:

ω(z) =

[(z − z0)

λ

]n. (4.41)

Actually, we can also admit isolated singularities, so that n can take both positiveor negative values. In order to understand the meaning of such a solution, let ustake n = 1, an try to see to which field n it corresponds. Since n is real, it will beconvenient to consider

|ω(z)|2 =1

λ2

[(x− x0)2 + (y − y0)2

]=( rλ

)2

. (4.42)

On the other hand, we have

|ω|2 =4

(1− n3)2

(n2

1 + n22

)= 4

(1 + n3)

(1− n3), (4.43)

such that

n3 =(|ω|/2)2 − 1

(|ω|/2)2 + 1=

(r/2λ)2 − 1

(r/2λ)2 + 1,

n1 =(1− n3)

2ω1 =

(x− x0)/λ

(r/2λ)2 + 1,

n2 =(1− n3)

2ω2 =

(y − y0)/λ

(r/2λ)2 + 1. (4.44)

Using polar coordinates with center in (x0, y0), we have finally

n =1

1 +(r

2λ

)2

[r

λcosϕ,

r

λsinϕ,

( r2λ

)2

− 1

], (4.45)


such that at r = 0 the spin is pointing down, at r →∞ the spins are pointing up, andλ gives the size of the soliton solution, that goes under the name of skyrmion, afterT. Skyrme, who saw such solitary solutions as a possible way to obtain nucleons.

It remains to see what happens for general n. In particular, we would like tosee, how such field configurations are connected to the different topological sectorsdescribed by Q. Since we are interested in the case where the fields satisfy the lowerbound, that is, when the equality in (4.27) holds, it is easier to calculate the energydensity instead of calculating directly Q. First we see how the energy density canbe related to ω. Let us consider∣∣∣∣dωdz

∣∣∣∣2 = (∂1ω1)2 + (∂1ω2)2

=4

(1− n3)4[∂µn1∂µn1 + ∂µn2∂µn2 + 2 (∂1n1∂2n2 − ∂2n1∂1n2)] ,(4.46)

where we used the expressions (4.40) with their lower signs, since we are dealingwith ω as a function of z. The last term in the square brackets can be rewrittenusing (4.35), as follows:

∂1n1∂2n2 − ∂2n1∂1n2 = ε3ab∂1na∂2nb = n3n · (∂1n× ∂2n) . (4.47)

Using (4.27) with the lower sign, we have

∂1n1∂2n2 − ∂2n1∂1n2 = −n3 ∂µn · ∂µn . (4.48)

On the other hand, using the relations (4.38) and the constraint of the field n, onecan show that

∂µn3∂µn3 =(1− n2

3)

2∂µn · ∂µn . (4.49)

Putting the results above together, one arrives at

E =

∫d2x

∣∣dωdz

∣∣2(1 + |ω|2/4)2

= n2λ2n

∫d2x

|z − z0|2(n−1)

(λ2n + |z − z0|2n/4)2 = 4πn ,(4.50)

where for the last step is more convenient to go over to polar coordinates centeredat z0. Then, we see that the powers give directly the different topological sectorsof the skyrmions. The result above also shows that E does not depend on λ, theextension of the skyrmion. In fact, the field configuration obtained for n = 1, eq.(4.45), stays invariant upon a change of λ and a corresponding rescaling of space.Furthermore, the energy (or the action) are scale invariant, as we already pointedout in Sec. 4.1.

4.2.3 Anyons in 2+1 dimensions

Once we have seen that in two dimensions the non-linear σ-model has topologicalnon-trivial configurations, we can ask, what happens if such configurations can travel


in time. For that let us add a third dimension as euclidean time. Here we followthe discussion by Wilczek and Zee [14].

The skyrmions we have discussed previously can be seen as field configurationsat a given time. Their density is the temporal part of a current

jµ =1

8πεµνλn · (∂νn× ∂λn) . (4.51)

But one sees immediately, that this is a conserved current since

∂µjµ = 0 . (4.52)

Being in 2+1 euclidean space, the conservation above can be written in the morefamiliar form ∇ · j = 0, such that there is a gauge field A fulfilling j = ∇×A, or

jµ = εµνλ∂νAλ . (4.53)

Having a gauge field, we can as usual have a term in the action connecting thecurrent to the gauge field:

H =

∫d3x jµAµ , (4.54)

and ask ourselves, if such a term makes sense.A first interesting property is that H is invariant under global O(3) rotations as

the original non-linear σ-model. To see this let us first consider the change in jµ bya rotation g ∈ O(3), that is, a change n′ = gn. For the current we have

jµ′ = εµνλεabcgaa′na′∂ν(gbb′n

b′)∂λ(gcc′nc′) . (4.55)

But

εabcgaa′gbb′gcc′ = det(g) εa′b′c′ , (4.56)

such that

jµ′ = det(g) jµ . (4.57)

Since jµ and Aµ are related by (4.53), we have also Aµ′ = det(g)Aµ, such that H

is O(3) invariant, and the action could in principle possess such a term. A secondproperty of H is its scale invariance. As shown in eq. (4.8), for d = 3, the couplinghas dimension g ∼ a, such that the fields n remain dimensionless. Then, we havethat jµ ∼ a−2, and consequently Aµ ∼ a−1, such that upon rescaling distances, Hremains invariant.

We consider now a general variation of the fields and see how H varies.

δH =

∫d3x (δjµAµ + jµδAµ)

=

∫d3x

(Aµε

µνλ∂νδAλ + jµδAµ)

= 2

∫d3x jµδAµ , (4.58)


where on going from the second to the third line, we integrated by parts setting thevariation to zero at the boundaries. To obtain the variation of the gauge-field, welook at

δjµ = εµνλ∂νδAλ , (4.59)

on the one hand, and on the other perform it directly,

δjµ =1

8πεµνλεabcδ

(na∂νn

b∂λnc)

=1

4πεµνλ∂ν

(εabcnaδnb∂λn

c), (4.60)

such that

δAµ =1

4πεabcnaδnb∂µn

c , (4.61)

leading to

jµδAµ =1

32π2εabcεdefnaδnbndεµνλ∂µn

c∂νne∂λn

f = 0 . (4.62)

To see this, notice that the product of derivatives does not vanish only when c, e, andf are all different from each other. But this implies that c = d. Now consider theexpression for a given value for b. There are 4 possible combinations for the parityof the permutations of (a, b, c) and (c, e, f). The contributions from (even,even) and(odd,odd) cancel each other, as well as those for (even,odd) and (odd,even). Thelast results shows that H is an homotopic invariant, such that a topological meaningof this quantity can be expected.

In order to understand H further, we consider the curl of jµ, that leads to

∂ν∂νAµ = −εµλρ∂λjρ , (4.63)

where we chose the ”Lorentz” gauge ∂µAµ = 0. Hence, solving Poisson’s equation

above, we obtain the gauge-field

Aµ(x) =1

4π

∫d3x′

εµλρ∂λ′jρ(x′)

|x− x′|

= − 1

4π

∫d3x′

εµλρ (x− x′)λ jρ(x′)|x− x′|3

, (4.64)

where in the last line we integrated by parts. Finally, for H we have

H = − 1

4π

∫d3x d3x′

εµλρjµ(x) (x− x′)λ jρ(x′)|x− x′|3

, (4.65)

Let us assume now that the currents above correspond to two skyrmions well sepa-rated from each other, and let us rescale space-time such that they are reduced totheir world-lines:

jµd3x→ Id`µ . (4.66)


Then, we can write

H = −II′

4π

∮Γ

∮Γ′

d` · [(x− x′)× d`′]

|x− x′|3, (4.67)

where we have assumed that the world-lines are closed curves, as is the case forperiodic boundary conditions in time. The expression above resembles Amperes lawin electrodynamics for the force between two cable loops with circulating currents.The big difference is that in the electrodynamic case we have a vector but here wehave a pseudoscalar.

In order to see the meaning of the integral above, let us assume that Γ and Γ′

do not intersect each other, i.e. they do not have any point in common, and rewrite(4.64) for the world-line,

A(x) = − I ′

4π

∮Γ′

(x− x′)× d`′

|x− x′|3. (4.68)

Here we can apply a modified form of Stokes’ theorem. In its original form one has∮Γ

M · d` =

∫F

(∇×M ) · df . (4.69)

We can take M = C×D, where D is a constant vector. Then, with d` = t d`, anddf = n df , one has

Di

∮Γ

(t×C

)id` = Di

∫F

[(∂Cj∂xi

)nj −

(∂Cj∂xj

)ni

]df , (4.70)

that holds for any D, such that finally,∮Γ

C × d` = −∫F

(n×∇)×C df . (4.71)

In our case we have

C =(x− x′)|x− x′|3

, (4.72)

such that going back to (4.68), we have

A(x) =I ′

4π

∫F

(n×∇)× (x− x′)|x− x′|3

df

=I ′

4π

∫F

[∂

∂x′i

(x− x′)j|x− x′|3

]nj −

[∂

∂x′j

(x− x′)j|x− x′|3

]ni

df . (4.73)

But since

∇′ ·(x− x′)j|x− x′|3

= 4′ 1

|x− x′|= −4πδ (x− x′) , (4.74)


and we are interested on x 6= x′, the second term above can be discarded, leadingto

A(x) = ∇ϕ(x) , (4.75)

with

ϕ(x) =I ′

4π

∫F

∇′ 1

|x− x′|· df

=I ′

4π

∫F

(x− x′)|x− x′|3

· df =I ′

4πΩ . (4.76)

Hence we see that the potential is just the solid angle subtended by the surfacespanned by the curve Γ′. In order to have H, we need to perform the other lineintegral.

H = I

∮Γ

d` ·∇ϕ = I ϕ|Γ (4.77)

Here we have to distinguish two cases. If the Γ and Γ′ are not linked, then eachintegral is performed around a simply connected region, such that on a closed loopϕ|Γ = 0. On the other hand, if the loops are linked, the region is not any moresimply connected. Going on a close loop we generate a closed surface, such that∮

Γ

∮Γ′

d` · [(x− x′)× d`′]

|x− x′|3= Ω|Γ = 4πn , (4.78)

where n is the number of times one curve winds around the other. This integral isdue to Gauß. To convince oneself that this is so, we can take Γ a line extending from−∞ to ∞ such that x = (0, 0, t1) with −∞ ≤ t1 ≤ ∞, and Γ′ a circle of radius onebeing pierced by Γ at the center, such that x′ = (cos t2, sin t2, 0), with 0 ≤ t2 ≤ 2π.Then, ∮

Γ

∮Γ′

d` · [(x− x′)× d`′]

|x− x′|3=

∫ ∞−∞

dt1

∫ 2π

0

dt2

(1 + t21)3/2

= 4π . (4.79)

We have shown that H is related to the linking number of the world-lines of theskyrmions. The form (4.54) together with (4.53) is also called a Chern-Simons term,in this case for a U(1) gauge theory. If the action happens to have a term iθH, sucha term in the exponential gives rise to a phase each time the skyrmions wind aroundeach other, determining the statistics of these objects. Since there is no constraintto the factor in front of H, any statistics is possible, giving rise to anyons.

Bibliography

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[6] A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems(McGraw-Hill, New York, 1971).

[7] A. Perelomov, Generalized Coherent States and Their Applications (SpringerVerlag, Berlin, 1986).

[8] M. V. Berry, Proc. R. Soc. Lond. A 392, 45 (1984).

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