Quantum phase transitions of correlated electrons and atoms See also: Quantum phase transitions of...

Quantum phase transitions

of correlated electrons and atoms

See also: Quantum phase transitions of correlated electrons in two dimensions,

cond-mat/0109419.

Quantum Phase Transitions Cambridge University Press

Subir SachdevHarvard University

What is a quantum phase transition ?

Non-analyticity in ground state properties as a function of some control parameter g

E

g

True level crossing:

Usually a first-order transition

E

g

Avoided level crossing which becomes sharp in the infinite

volume limit:

second-order transition

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.

~z

cg g

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

characteristic energy scale at other values of g:

• 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.

Outline

I. Quantum Ising Chain

II. Landau-Ginzburg-Wilson theory Mean field theory and the evolution of the

excitation spectrum.

III. Superfluid-insulator transitionBoson Hubbard model at integer filling.

IV. Bosons at fractional fillingBeyond the Landau-Ginzburg-Wilson paradigm.

V. Quantum phase transitions and the Luttinger theoremDepleting the Bose-Einstein condensate of

trapped ultracold atoms – see talk by Stephen Powell


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:

xj

j

H Jg 2Jg

j

j

1 1

Coupling between qubits:

z zj j

j

H J

1 1

Prefers neighboring qubits

are

(not entangle

d)

j j j jeither or

1 1j j j j

0 1 1x z zj j j

j

J gH H H Full Hamiltonian

leads to entangled states at g of order unity

LiHoF4

Experimental realization

Weakly-coupled qubits Ground 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

j

p e

2 1

1

Excitation energy 4 sin2

Excitation gap 2 2

pap J O g

gJ J O g

p

a

a

p

1g

Dynamic Structure Factor :

Cross-section to flip a to a (or vice versa)

while transferring energy

and momentum

( , )

p

S p

,S p Z p

Three quasiparticlecontinuum

Quasiparticle pole

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

phase coherence time

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

~3

Weakly-coupled qubits 1g

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

p

a

a

p

Second state obtained by

and mix only at order N

G

G G g

0

Ferromagnetic moment

0zN G G

Strongly-coupled qubits 1g




and momentum

( , )

p

S p

,S p 22

0 2N p

Two domain-wall continuum

Structure holds to all orders in g

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

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

T

k Te

phase coherence time

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

~2

Strongly-coupled qubits 1g

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

1/8

0 ~ cN g g

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

ggc

Excitation energy gap ~ cg g




and momentum

( , )

p

S p

,S p

Critical coupling cg g

c p

7 /82 2 2~ c p

No quasiparticles --- dissipative critical continuum

Quasiclassical dynamics


1/ 4

1~z z

j kj k

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

i

zi

zi

xiI gJH 1

0

7 / 4

( ) , 0

1 / ...

2 tan16

z z i tj k

k

R

BR

idt t e

A

T i

k T

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

Outline








II. Landau-Ginzburg-Wilson theory

Mean field theory and the evolution of the excitation spectrum

Outline








III. Superfluid-insulator transition

Boson Hubbard model at integer filling

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

Bose condensationVelocity distribution function of ultracold 87Rb atoms

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=7ErV0=10Er

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

V0=3Er

Bosons at filling fraction f

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

Weak interactions: superfluidity

Strong interactions: Mott insulator which preserves all lattice

symmetries



0

Strong interactions: insulator


0

The Superfluid-Insulator transition

†

†

Degrees of freedom: Bosons, , hopping between the

sites, , of a lattice, with short-range repulsive interactions.

- ( 1)2

j

i j jj j

ji j

j

b

b b n n

j

UnH t

† j j jn b b

Boson Hubbard model

For small U/t, ground state is a superfluid BEC with

superfluid density density of bosons

M.PA. Fisher, P.B. Weichmann, G. Grinstein, and D.S. Fisher

Phys. Rev. B 40, 546 (1989).

What is the ground state for large U/t ?

Typically, the ground state remains a superfluid, but with

superfluid density density of bosons

The superfluid density evolves smoothly from large values at small U/t, to small values at large U/t, and there is no quantum phase transition at any intermediate value of U/t.

(In systems with Galilean invariance and at zero temperature, superfluid density=density of bosons always, independent of the strength of the interactions)

What is the ground state for large U/t ?

Incompressible, insulating ground states, with zero superfluid density, appear at special commensurate densities

3jn t

U

7 / 2jn Ground state has “density wave” order, which spontaneously breaks lattice symmetries

Excitations of the insulator: infinitely long-lived, finite energy quasiparticles and quasiholes

2

, , *,

Energy of quasi-particles/holes: 2p h p h

p h

pp

m

Boson Green's function :

Cross-section to add a boson

while transferring energy and momentum

( , )

p

G p

Continuum of two quasiparticles +

one quasihole

,G p Z p

Quasiparticle pole

~3

Insulating ground state

Similar result for quasi-hole excitations obtained by removing a boson

Entangled states at of order unity

ggc

Quasiparticle weight Z

~ cZ g g

ggc

Superfluid density s

( 2)~

d z

s cg g

gc

g

Excitation energy gap

,

,

~ for

=0 for

p h c c

p h c

g g g g

g g

/g t U

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



Relaxational dynamics ("Bose molasses") with

phase coherence/relaxation time given by

1 Universal number 1 K 20.9kHzBk T

S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 2411 (1992).K. Damle and S. SachdevPhys. Rev. B 56, 8714 (1997).

Crossovers at nonzero temperature

2

Conductivity (in d=2) = universal functionB

Q

h k T

M.P.A. Fisher, G. Girvin, and G. Grinstein, Phys. Rev. Lett. 64, 587 (1990).K. Damle and S. Sachdev Phys. Rev. B 56, 8714 (1997).

Outline








IV. Bosons at fractional filling

Beyond the Landau-Ginzburg-Wilson paradigm

L. Balents, L. Bartosch, A. Burkov, S. Sachdev, K. Sengupta, Physical Review B 71, 144508 and 144509 (2005),

cond-mat/0502002, and cond-mat/0504692.



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)

0




0

Strong interactions: insulator



0


1

2( + )

.Can define a common CDW/VBS order using a generalized "density" ie Q rQ

Q

r

Insulating phases of bosons at filling fraction f

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

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


All insulators have 0 and 0 for certain Q Q


1

2( + )


Q

r








Q

r






1

2( + )


1

2( + )


Q

r






Ginzburg-Landau-Wilson approach to multiple order parameters:

charge int

2 4

1 1

2 4

charge 2 2

22

int

sc sc

sc sc sc sc

sc

F F F F

F r u

F r u

F v

Q

Q Q Q

Q

Distinct symmetries of order parameters permit couplings only between their energy densities

S. Sachdev and E. Demler, Phys. Rev. B 69, 144504 (2004).

Predictions of LGW theory

(Supersolid)

Coexistence

1 2r r

First order transitionsc

1 2r rSuperconductor

Superconductor Charge-ordered insulator

sc Q

Q

Charge-ordered insulator

1 2r rSuperconductor

sc Q( topologically ordered)

" "

0, 0sc

Disordered

Q

Charge-ordered

insulator

Excitations of the superfluid: Vortices

J. R. Abo-Shaeer, C. Raman, J. M. Vogels, and W. Ketterle, Observation of Vortex Lattices in Bose-Einstein Condensates, Science 292, 476 (2001).

Observation of quantized vortices in rotating ultracold Na

Quantized fluxoids in YBa2Cu3O6+y

J. C. Wynn, D. A. Bonn, B.W. Gardner, Yu-Ju Lin, Ruixing Liang, W. N. Hardy, J. R. Kirtley, and K. A. Moler, Phys. Rev. Lett. 87, 197002 (2001).

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” ?

Excitations of the superfluid: Vortices

In ordinary fluids, vortices experience the Magnus Force

FM

mass density of air velocity of ball circulationMF

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)

A4

A3

A1

A2 A1+A2+A3+A4= 2f

where f is the boson filling fraction.


• At f=1, the “magnetic” flux per unit cell is 2, 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.

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

, : 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

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

Space group symmetries of Hofstadter Hamiltonian:

Bosons at rational filling fraction f=p/q

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 in

the vortex spectrum. These vortices realize

the smallest, -dimensional, representation of

the

f p q q

q

q

q

magnetic algebra.

At filling = / , there are species

of vortices, (with =1 ),

associated with degenerate minima in

the vortex spectrum. These vortices realize

the smallest, -dimensional, representation of

the

f p q q

q

q

q

magnetic algebra.

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

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

21

2

1

: ; :

1 :

i fx y

qi mf

mm

T T e

R eq

Superconductor insulator : 0 0


The vortices characterize

superconducting and VBS/CDW orders

q both The vortices characterize


q both

The vortices characterize


q both The vortices characterize


q both

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

pm

e

nq

T e T e

e

Q

Q QQ Q Q Q

Q

ˆ

:

y

R R 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

q

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

1

2( + )

Spatial structure of insulators for q=4 (f=1/4 or 3/4)

unit cells;

, , ,

all integers

a b

q q aba b q

unit cells;

, , ,

all integers

a b

q q aba b q

Field theory with projective symmetry






q

lies a







q

lies a


Any pinned vortex must pick an orientation in flavor

space: this induces a halo of VBS order in its vicinity

100Å

b7 pA

0 pA

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

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).

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).

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

10

1 THz = 4 meV

etect vortex zero point motion

by

v e

Preliminar

m m

y

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/q per site, while the density of the superfluid is close (but need

not be identical) to this value

Quantum phase transitions of correlated electrons and atoms See also: Quantum phase transitions of...

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