Subir Sachdev

Quantum phase transitions of

ultracold atoms

Transparencies online at http://pantheon.yale.edu/~subir

Quantum Phase TransitionsCambridge University Press (1999)

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.

~ zcg g ν∆ −

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. The superfluid—Mott-insulator transition

II. Mott insulator in a strong electric field.

III. Conclusions

I. The superfluid—Mott-insulator transition

S. Sachdev, K. Sengupta, and S. M. Girvin, Physical Review B 66, 075128 (2002).

I. 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

jU nH t µ= − + − +∑ ∑ ∑

† 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 = tU

−

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

ppm

ε = ∆ +

Boson Green's function : Cross-section to add a boson while transferring energy and momentum

( , )

p

G p

ω

ω

Continuum of two quasiparticles +

one quasihole

ω

( ),G p ω( )( )Z pδ ω ε−

Quasiparticle pole

~3∆

Insulating ground state

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

Entangled states at of order unity

ggc

Quasiparticleweight Z

( )~ cZ g g ην−

ggc

Superfluiddensity ρs

( )( 2)~ d zs cg g νρ + −−

gc

g

Excitation energy gap ∆

( ),

,

~ for

=0 for p h c c

p h c

g g g g

g g

ν∆ − <

∆ >

/g t U≡

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

Quasiclassicaldynamics

Quasiclassicaldynamics

( ) ( )

Relaxational dynamics ("Bose molasses") with phase coherence/relaxation time given by

1 Universal number 1 K 20.9kHzBk Tϕ

ϕ

τ

µτ

= =

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

Qh k T

ω ∑ ∑→

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

I. The superfluid—Mott-insulator transition

II. Mott insulator in a strong electric field.

III. Conclusions

II. Mott insulator in a strong electric fieldS. Sachdev, K. Sengupta, and S. M. Girvin, Physical Review B 66, 075128 (2002).

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

Related earlier work by C. Orzel, A.K. Tuchman, M. L. Fenselau, M. Yasuda, and M. A. Kasevich, Science 291, 2386 (2001).

Superfluid-insulator transition of 87Rb atoms in a magnetic trap and an optical lattice potential

Detection method

Trap is released and atoms expand to a distance far larger than original trap dimension

( ) ( ) ( )2

, exp ,0 exp ,02 2

where with = the expansion distance, and position within trap

m mmT i i iT T T

ψ ψ ψ ⋅

= ≈ +

=

0 020 0

0 0

R R rRR

R = R + r, R r

In tight-binding model of lattice bosons bi ,

detection probability ( )( )† 0

,exp with i j i j

i j

mb b i

T∝ ⋅ − =∑ Rq r r q

Measurement of momentum distribution function

Schematic three-dimensional interference pattern with measured absorption images taken along two orthogonal directions. The absorption images were obtained after ballistic

expansion from a lattice with a potential depth of V0 = 10 Er and a time of flight of 15 ms.

Superfluid state

Superfluid-insulator transition

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

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

V0=3Er

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

Applying an “electric” field to the Mott insulator

V0=10 Erecoil τperturb = 2 ms V0= 13 Erecoil τperturb = 4 ms

V0= 16 Erecoil τperturb = 9 ms V0= 20 Erecoil τperturb = 20 ms

What is the quantum state

here ?

( ) ( )† †

†

12

i j j i i i i iij i i

i i i

UH t b b b b n n n

n b b

= − + + − − ⋅

=

∑ ∑ ∑E r

, ,U E t E U−

Describe spectrum in subspace of states resonantly coupled to the Mott insulator

Effective Hamiltonian for a quasiparticle in one dimension (similar for a quasihole):

( )† † †eff 1 13 j j j j j j

jH t b b b b Ejb b+ +

= − + + ∑

( ) ( )Exact eigenvalues ;Exact eigenvectors 6 /

m

m j m

Em mj J t E

ε

ψ −

= = −∞ ∞

=

All charged excitations are strongly localized in the plane perpendicular electric field.Wavefunction is periodic in time, with period h/E (Bloch oscillations)

Quasiparticles and quasiholes are not accelerated out to infinity

dk Edt

=

Semiclassical picture

k

Free particle is accelerated out to infinityIn a weak periodic potential, escape to infinity occurs via Zenertunneling across band gaps

Experimental situation: Strong periodic potential in which there is negligible Zener tunneling, and the

particle undergoes Bloch oscillations

Creating dipoles on nearest neighbor links creates a state with relative energy U-2E ; such states are not

part of the resonant manifold

Nearest-neighbor dipole

Important neutral excitations (in one dimension)

Nearest-neighbor dipoles

Dipoles can appear resonantly on non-nearest-neighbor links.Within resonant manifold, dipoles have infinite on-link

and nearest-link repulsion

Nearest neighbor dipole

A non-dipole state

State has energy 3(U-E) but is connected to resonant state by a matrix element smaller than t2/U

State is not part of resonant manifold

Hamiltonian for resonant dipole states (in one dimension)

( )

†

† †

† † †1 1

Creates dipole on link

6 ( )

Constraints: 1 ; 0

d

d

H t d d U E d d

d d d d d d+ +

⇒

= − + + −

≤ =

∑ ∑

Note: there is no explicit dipole hopping term.

However, dipole hopping is generated by the interplay of terms in Hd and the constraints.

Determine phase diagram of Hd as a function of (U-E)/t

Weak electric fields: (U-E) t

Ground state is dipole vacuum (Mott insulator)

† 0dFirst excited levels: single dipole states

0

Effective hopping between dipole states † 0d

† † 0md d† 0md

0

If both processes are permitted, they exactly cancel each other.The top processes is blocked when are nearest neighbors

2

A nearest-neighbor dipole hopping term ~ is generatedtU E−

⇒

,m

t t

t t

Strong electric fields: (E-U) t

Ground state has maximal dipole number.

Two-fold degeneracy associated with Ising density wave order:† † † † † † † † † † † †1 3 5 7 9 11 2 4 6 8 10 120 0d d d d d d or d d d d d d

Ising quantum critical point at E-U=1.08 t

-1.90 -1.88 -1.86 -1.84 -1.82 -1.800.200

0.204

0.208

0.212

0.216

0.220

πS /N3/4

λ

N=8 N=10 N=12 N=14 N=16

Equal-time structure factor for Ising order

parameter

Hamiltonian for resonant states in higher dimensions

( )

( )

( )

† †1, , 1, ,

,

† †,

† † † †, , , , , , , ,

, , ,,

† †, , , ,

,

6

( ) 2

2 3 H.c.

1 ; 1 ; 0

ph n n n nn

n n n nn

n m

n n n n n n n

n mnm

n

H t p h p h

U E p p h h

t h h p

p p h h p p h h

p

+ += − +

≤ ≤ =

−+ +

− + +

∑

∑

∑

†,

†,

Creates quasiparticle in column and transverse position

Creates quasihole in column and transverse position n

n

p n

h n

⇒

⇒

Terms as in one dimension

Transverse hopping

Constraints

New possibility: superfluidity in transverse direction (a smectic)

Resonant states in higher dimensionsResonant states in higher dimensionsQuasiparticles

Quasiholes

Dipole states in one dimension

Quasiparticles and quasiholes can move

resonantly in the transverse directions in higher

dimensions.

Constraint: number of quasiparticles in any column = number of quasiholes in column to its left.

Ising density wave order

( ) /U E t= −

( ) /U E t= −

Transverse superfluidity

Possible phase diagrams in higher dimensions

Implications for experiments

•Observed resonant response is due to gapless spectrum near quantum critical point(s).

•Transverse superfluidity (smectic order) can be detected by looking for “Bragg lines” in momentum distribution function---bosons are phase coherent in the transverse direction.

•Present experiments are insensitive to Ising density wave order. Future experiments could introduce a phase-locked subharmonicstanding wave at half the wave vector of the optical lattice---this would couple linearly to the Ising order parameter.

Conclusions

I. Study of quantum phase transitions offers a controlled and systematic method of understanding many-body systems in a region of strong entanglement.

II. Atomic gases offer many exciting opportunities to study quantum phase transitions because of ease by which system parameters can be continuously tuned.