quantum phase transitions of correlated electrons …qpt.physics.harvard.edu/talks/tobago.pdfquantum...

Quantum phase transitions of correlated electrons and atoms

Physical Review B 71, 144508 and 144509 (2005),cond-mat/0502002

Leon Balents (UCSB) Lorenz Bartosch (Yale) Anton Burkov (UCSB) Subir Sachdev (Yale)

Krishnendu Sengupta (Toronto)

T Quantum-critical

Why study quantum phase transitions ?

ggc• Theory for a quantum system with strong correlations: describe phases on either side of gc by expanding in deviation from the quantum critical point. • Critical point is a novel state of matter without quasiparticle excitations

• Critical excitations control dynamics in the wide quantum-critical region at non-zero temperatures.

~ zcg g ν∆ −

Important property of ground state at g=gc : temporal and spatial scale invariance;

characteristic energy scale at other values of g:

OutlineOutlineI. The Quantum Ising chain

II. The superfluid-Mott insulator quantum phase transition

III. The cuprate superconductorsSuperfluids proximate to finite doping Mott insulators with VBS order ?

IV. Vortices in the superfluid

V. Vortices in superfluids near the superfluid-insulator quantum phase transition

The “quantum order” of the superconducting state: evidence for vortex flavors

I. Quantum Ising Chain

I. Quantum Ising Chain

( ) ( )

Degrees of freedom: 1 qubits, "large"

,

1 1 or , 2 2

j j

j jj j j j

j N N=

=→

↓

↓

↑

↑ ↑+ =← − ↓

…

0

Hamiltonian of decoupled qubits: x

jj

H Jg σ= − ∑ 2Jg

j→

j←

1 1

Coupling between qubits: z z

j jj

H J σ σ += − ∑

( )( )1 1j j j j+ ++ +← ←→ ←→ →← →

1 1

Prefers neighboring qubits

are

(not entangle

d)j j j j

either or+ +

↓ ↓↑ ↑

( )0 1 1x z zj j j

jJ gH H H σ σ σ += + = − +∑

Full Hamiltonian

leads to entangled states at g of order unity

LiHoF4

Experimental realization

Weakly-coupled qubitsGround state:

1 2

G

g

→→→→→→→→→→

→→→→ →→→

=

−←←− →

Lowest excited states:

jj →→→→ +→←= →→→→

Coupling between qubits creates “flipped-spin” quasiparticle states at momentum p

Entire spectrum can be constructed out of multi-quasiparticle states

jipxj

jp e= ∑

( ) ( )

( )

2 1

1

Excitation energy 4 sin2

Excitation gap 2 2

pap J O g

gJ J O g

ε −

−

⎛ ⎞= ∆ + +⎜ ⎟⎝ ⎠

∆ = − + p

∆

aπ

aπ

−

( )pε

( )1g

Dynamic Structure Factor : Cross-section to flip a to a (or vice versa) while transferring energy

and momentum

( , )

p

S p

ω

ω→ ←

ω

( ),S p ω( )( )Z pδ ω ε−

Three quasiparticlecontinuum

Quasiparticle pole

~3∆

Weakly-coupled qubits ( )1g

Structure holds to all orders in 1/g

At 0, collisions between quasiparticles broaden pole to a Lorentzian of width 1 where the

21is given by

Bk TB

T

k Te

ϕ ϕ

ϕ

τ τ

τ π−∆

>

=

phase coherence time

S. Sachdev and A.P. Young, Phys. Rev. Lett. 78, 2220 (1997)

Ground states:

2

G

g

=

−

↑ ↑↑↑↑↑↑↑↑↑↑

↑↑↑↑ −↑↑↓↑↑↑

Lowest excited states: domain walls

jjd ↓↓↑ ↓↑↑ ↓ +↑↑= ↓Coupling between qubits creates new “domain-

wall” quasiparticle states at momentum pjipx

jj

p e d= ∑( ) ( )

( )

2 2

2

Excitation energy 4 sin2

Excitation gap 2 2

pap Jg O g

J gJ O g

ε ⎛ ⎞= ∆ + +⎜ ⎟⎝ ⎠

∆ = − +

p

∆

aπ

aπ

−

( )pε

Second state obtained by

and mix only at order N

G

G G g

↓ ↓

↑↓

⇔↑ 0

Ferromagnetic moment0zN G Gσ= ≠

Strongly-coupled qubits ( )1g


and momentum

( , )

p

S p

ω

ω→ ←

Strongly-coupled qubits ( )1g

ω

( ),S p ω( ) ( ) ( )22

0 2N pπ δ ω δ

Two domain-wall continuum

Structure holds to all orders in g~2∆

At 0, motion of domain walls leads to a finite ,

21and broadens coherent peak to a width 1 where Bk TB

T

k Te

ϕ

ϕϕ

τ

ττ π

−∆

>

=

phase coherence time

S. Sachdev and A.P. Young, Phys. Rev. Lett. 78, 2220 (1997)

Entangled states at g of order unity

ggc

“Flipped-spin” Quasiparticle

weight Z

( )1/ 4~ cZ g g−

A.V. Chubukov, S. Sachdev, and J.Ye, Phys. Rev. B 49, 11919 (1994)

ggc

Ferromagnetic moment N0

( )1/80 ~ cN g g−

P. Pfeuty Annals of Physics, 57, 79 (1970)

ggc

Excitation energy gap ∆ ~ cg g∆ −


and momentum

( , )

p

S p

ω

ω→ ←

Critical coupling ( )cg g=

ω

( ),S p ω

c p

( ) 7 /82 2 2~ c pω−

−

No quasiparticles --- dissipative critical continuum

Quasiclassicaldynamics

Quasiclassicaldynamics

1/ 41~z z

j kj k

σ σ−

P. Pfeuty Annals of Physics, 57, 79 (1970)

( )∑ ++−=i

zi

zi

xiI gJH 1σσσ

( ) ( )

( )

0

7 / 4

( ) , 0

1 / ...

2 tan16

z z i tj k

k

R

BR

i dt t e

AT i

k T

ωχ ω σ σ

ω

π

∞

⎡ ⎤= ⎣ ⎦

=− Γ +

⎛ ⎞Γ = ⎜ ⎟⎝ ⎠

∑∫

S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 2411 (1992).S. Sachdev and A.P. Young, Phys. Rev. Lett. 78, 2220 (1997).

II. The superfluid-Mott insulator quantum phase transition

Bose condensationVelocity distribution function of ultracold 87Rb atoms

M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wiemanand E. A. Cornell, Science 269, 198 (1995)

Apply a periodic potential (standing laser beams) to trapped ultracold bosons (87Rb)

Momentum distribution function of bosons

Bragg reflections of condensate at reciprocal lattice vectors

M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002).

Superfluid-insulator quantum phase transition at T=0

V0=0Er V0=7Er V0=10Er

V0=13Er V0=14Er V0=16Er V0=20Er

V0=3Er

Bosons at filling fraction f = 1Weak interactions:

superfluidity

Strong interactions: Mott insulator which preserves all lattice

symmetries

M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002).

Bosons at filling fraction f = 1

0Ψ ≠

Weak interactions: superfluidity


0Ψ =

Strong interactions: insulator

Bosons at filling fraction f = 1/2

0Ψ ≠

Weak interactions: superfluidity

C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, 134510 (2001) S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)

Bosons at filling fraction f = 1/2

0Ψ =

Strong interactions: insulator


12( + )=

Insulating phases of bosons at filling fraction f = 1/2

Valence bond Valence bond solid (VBS) ordersolid (VBS) order


Charge density Charge density wave (CDW) orderwave (CDW) order

( ) .Can define a common CDW/VBS order using a generalized "density" ieρ ρ= ∑ Q rQ

Q

r

All insulators have 0 and 0 for certain ρΨ = ≠Q Q



12( + )=





Q

r



12( + )=






Q

r



12( + )=



Q

r






12( + )=






Q

r



Superfluid-insulator transition of bosons at generic filling fraction f

The transition is characterized by multiple distinct order parameters (boson condensate, VBS/CDW order)

Traditional (Landau-Ginzburg-Wilson) view:Such a transition is first order, and there are no precursor fluctuations of the order of the insulator in the superfluid.

Superfluid-insulator transition of bosons at generic filling fraction f

The transition is characterized by multiple distinct order parameters (boson condensate, VBS/CDW order)

Traditional (Landau-Ginzburg-Wilson) view:Such a transition is first order, and there are no precursor fluctuations of the order of the insulator in the superfluid.

Recent theories:Quantum interference effects can render such transitions second order, and the superfluid does contain precursor VBS/CDW fluctuations.

N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989).T. Senthil, A. Vishwanath, L. Balents, S. Sachdev and M.P.A. Fisher, Science 303, 1490 (2004).

III. The cuprate superconductors

Superfluids proximate to finite doping Mott insulators with VBS order ?

Cu

O

LaLa2CuO4

La2CuO4

Mott insulator: square lattice antiferromagnet

jiij

ij SSJH ⋅= ∑><

La2-δSrδCuO4

Superfluid: condensate of paired holes

0S =

Many experiments on the cupratesuperconductors show:

• Tendency to produce modulations in spin singlet observables at wavevectors (2π/a)(1/4,0) and (2π/a)(0,1/4).

• Proximity to a Mott insulator at hole density δ =1/8 with long-range charge modulations at wavevectors(2π/a)(1/4,0) and (2π/a)(0,1/4).

The cuprate superconductor Ca2-xNaxCuO2Cl2

T. Hanaguri, C. Lupien, Y. Kohsaka, D.-H. Lee, M. Azuma, M. Takano, H. Takagi, and J. C. Davis, Nature 430, 1001 (2004).




Superfluids proximate to finite doping Mott insulators with VBS order ?

Experiments on the cuprate superconductors also show strong vortex fluctuations above Tc

Measurements of Nernsteffect are well explained by a model of a liquid of vortices and anti-vortices

N. P. Ong, Y. Wang, S. Ono, Y. Ando, and S. Uchida, Annalender Physik 13, 9 (2004).

Y. Wang, S. Ono, Y. Onose, G. Gu, Y. Ando, Y. Tokura, S. Uchida, and N. P. Ong, Science299, 86 (2003).

Main claims:

• There are precursor fluctuations of VBS order in the superfluid.

• There fluctuations are intimately tied to the quantum theory of vortices in the superfluid

IV. Vortices in the superfluid

Magnus forces, duality, and point vortices as dual “electric” charges

Excitations of the superfluid: Vortices

Central question:In two dimensions, we can view the vortices as

point particle excitations of the superfluid. What is the quantum mechanics of these “particles” ?

In ordinary fluids, vortices experience the Magnus Force

FM

( ) ( ) ( )mass density of air velocity of ball circulationMF = i i

Dual picture:The vortex is a quantum particle with dual “electric”

charge n, moving in a dual “magnetic” field of strength = h×(number density of Bose particles)

V. Vortices in superfluids near the superfluid-insulator quantum phase transition

The “quantum order” of the superconducting state:

evidence for vortex flavors

A3

A1+A2+A3+A4= 2π fwhere f is the boson filling fraction.

A2A4

A1


• At f=1, the “magnetic” flux per unit cell is 2π, and the vortex does not pick up any phase from the boson density.

• The effective dual “magnetic” field acting on the vortex is zero, and the corresponding component of the Magnus force vanishes.

Bosons at rational filling fraction f=p/q

Quantum mechanics of the vortex “particle” in a periodic potential with f flux quanta per unit cell

Space group symmetries of Hofstadter Hamiltonian:

, : Translations by a lattice spacing in the , directions

: Rotation by 90 degrees.x yT T x y

R

2

1 1 1 4

Magnetic space group: ;

; ; 1

ifx y y x

y x x y

T T e T T

R T R T R T R T R

π

− − −

=

= = =

The low energy vortex states must form a representation of this algebra

At filling = / , there are species of vortices, (with =1 ), associated with degenerate minima inthe vortex spectrum. These vortices realizethe smallest, -dimensional, representation of the

f p q qq

q

q

ϕ …

magnetic algebra.

Hofstadter spectrum of the quantum vortex “particle” with field operator ϕ

Vortices in a superfluid near a Mott insulator at filling f=p/q

21

2

1

: ; :

1 :

i fx y

qi mf

mm

T T e

R eq

π

π

ϕ ϕ ϕ ϕ

ϕ ϕ

+

=

→ →

→ ∑


The vortices characterize superconducting and VBS/CDW orders

q bothϕ

( )

ˆ

* 2

1

VBS order: Status of space group symmetry determined by

2density operators at wavevectors ,

: ;

:

qi mnf i mf

m

mn

n n

i x ix y

p m

e

nq

T e T e

e π π

πρ

ρ ρ ρ

ρ ϕ ϕ

ρ

+=

=

=

→ →

∑i i

Q

Q QQ Q Q Q

Q

( ) ( )

ˆ

:

y

R Rρ ρ→Q Q


The excitations of the superfluid are described by the quantum mechanics of flavors of low energy vortices moving in zero dual "magnetic" field.

The orientation of the vortex in flavor space imp

qi

i lies a particular configuration of VBS order in its vicinity.

Spatial structure of insulators for q=2 (f=1/2)

Mott insulators obtained by “condensing” vortices

12( + )=

Spatial structure of insulators for q=4 (f=1/4 or 3/4)Field theory with projective symmetry

unit cells;

, , ,

all integers

a bq q ab

a b q

×




qi





qi


Any pinned vortex must pick an orientation in flavor space: this induces a halo of VBS order in its vicinityi

Vortex-induced LDOS of Bi2Sr2CaCu2O8+δ integrated from 1meV to 12meV at 4K

100Å

b7 pA

0 pA

Vortices have halos with LDOS modulations at a period ≈ 4 lattice spacings

Prediction of VBS order near vortices: K. Park and S. Sachdev, Phys.

Rev. B 64, 184510 (2001).

J. Hoffman, E. W. Hudson, K. M. Lang, V. Madhavan, S. H. Pan, H. Eisaki, S. Uchida, and J. C. Davis, Science 295, 466 (2002).

Measuring the inertial mass of a vortex

Measuring the inertial mass of a vortex

p

estimates for the BSCCO experiment:

Inertial vortex mass Vortex magnetoplasmon frequency

Future experiments can directly d

101 THz = 4 meV

etect vortex zero point motionby

v e

Preliminar

m m

y

ν≈

≈

looking for resonant absorption at this frequency.

Vortex oscillations can also modify the electronic density of states.

Superfluids near Mott insulators

• Vortices with flux h/(2e) come in multiple (usually q) “flavors”

• The lattice space group acts in a projective representation on the vortex flavor space.

• These flavor quantum numbers provide a distinction between superfluids: they constitute a “quantum order”

• Any pinned vortex must chose an orientation in flavor space. This necessarily leads to modulations in the local density of states over the spatial region where the vortex executes its quantum zero point motion.

Superfluids near Mott insulators

• Vortices with flux h/(2e) come in multiple (usually q) “flavors”

• The lattice space group acts in a projective representation on the vortex flavor space.

• These flavor quantum numbers provide a distinction between superfluids: they constitute a “quantum order”

• Any pinned vortex must chose an orientation in flavor space. This necessarily leads to modulations in the local density of states over the spatial region where the vortex executes its quantum zero point motion.

The Mott insulator has average Cooper pair density, f = p/qper site, while the density of the superfluid is close (but need

not be identical) to this value

quantum phase transitions of correlated electrons …qpt.physics.harvard.edu/talks/tobago.pdfquantum...

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