( ) ( )† † †1 ; ; Vary 2ij i j j i i i i i i

i j i

UH w c c c c n n n c c U wα α α α α α<

= − + + − =� �

A. S=1/2 fermions (Si:P , 2DEG)Metal-insulator transitionEvolution of magnetism across transition.Phil. Trans. Roy. Soc. A 356, 173 (1998) (cond-mat/9705074)Pramana 58, 285 (2002) (cond-mat/0109309)

B. S=0 bosons (ultracold atoms in an optical lattice)Superfluid-insulator transitionMott insulator in a strong electric field -S. Sachdev, K. Sengupta, S.M. Girvin, cond-mat/0205169

Ea U≈

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

ReznikovSarachik

Transport in coupled quantum dots150000 dots

A. S=1/2 fermionsU w small ; charge transport “metallic”

Magnetic properties of a single impurity

t

w

LT χ

LT χ

Low temperature magnetism dominated by such impurities

M. Milovanovic, S. Sachdev, R.N. Bhatt, Phys. Rev. Lett. 63, 82 (1989).R.N. Bhatt and D.S. Fisher, Phys. Rev. Lett. 68, 3072 (1992).

t/w

U/w

A. S=1/2 fermionsU w large ; charge transport “insulating”

R.N. Bhatt and P.A. Lee, Phys. Rev. Lett. 48, 344 (1982).B. Bernu, L. Candido, and D.M. Ceperley, Phys. Rev. Lett. 86, 870 (2002).G. Misguich, B. Bernu, C. Lhuillier, and C. Waldtmann, Phys. Rev. Lett. 81, 1098 (1998).

( ); ~ exp /ij i j ij i ji j

H J S S J a<

= ⋅ − −�� �

r r

Strong disorderRandom singlet phase

( )12

= ↑↓ − ↓↑Spins pair up in singlets

( ) ~ ~ ; 0.8P J J Tα αχ α− −� ≈

Weak disorderSpin gap

( )~ exp Tχ ∆−

Spin glass order

0iS ≠�

Higher density of moments + longer range of exchange interaction induces spin-glass order. Also suggested by strong-coupling flow of triplet interaction amplitude in Finkelstein’s (Z. Phys. B 56, 189 (1984)) renormalized weak-disorder expansion.

S. Sachdev, Phil. Trans. Roy. Soc. 356A, 173 (1998) (cond-mat/9705074); S. Sachdev, Pramana. 58, 285 (2002) (cond-mat/0109309).

U w

“Metallic”Local moments

“Insulating”Random singlets

/spin gap

Metal-insulator transition

Theory for spin glass transition in metalS. Sachdev, N. Read, R. Oppermann, Phys. Rev B 52, 10286 (1995).A.M. Sengupta and A. Georges, Phys. Rev B 52, 10295 (1995).

A. S=1/2 fermions

Glassy behavior observed by S. Bogdanovich and D. Popovic, cond-mat/0106545.

Effect of a parallel magnetic field on spin-glass state.

B

M

Singular behavior at a critical field at T=0Possibly related to observations of

S. A. Vitkalov, H. Zheng, K. M. Mertes, M. P. Sarachik, and T. M. Klapwijk, Phys. Rev. Lett. 87, 086401 (2001).

S. Sachdev, Pramana. 58, 285 (2002) (cond-mat/0109309).N. Read, S. Sachdev, and J. Ye, Phys. Rev. B 52, 384 (1995);

Spin glass order in plane orthogonal to B�

No order

Bσ

Mσ /d z B BM M TT

σσ ϕ −� �− = � �

� �

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

B. S=0 bosons

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 0� � �

20 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 ji 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

Coupled quantum dots !

V0=10 Ere coil ττττperturb = 2 m s V0= 13 Ere coil ττττperturb = 4 m s

V0= 16 Ere coil ττττperturb = 9 m s V0= 20 Ere coil ττττperturb = 20 m s

What is the quantum state

here ?

( ) ( )† †

†

12

i j j i i i i iij i i

i i i

UH w b b b b n n n

n b b

= − + + − − ⋅

=

� � �E r

, ,U E w 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 w b b b b Ejb b+ +

� �= − + +� ��

( ) ( )Exact eigenvalues ;Exact eigenvectors 6 /

m

m j m

Em mj J w E

εψ −

= = −∞ ∞=

�

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 Zener tunneling 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 w2/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 w 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)/w

Weak electric fields: (U-E) w

Ground state is dipole vacuum (Mott insulator)

�

† 0d�First excited levels: single dipole states

0

Effective hopping between dipole states † 0d�

† † 0md d�

† 0md0

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 generatedwU E−

�

, m�

w w

w w

Strong electric fields: (E-U) w�

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 w

-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

Resonant states in higher dimensionsResonant states in higher dimensions

Quasiparticles

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.

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 w p h p h

U E p p h h

w 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)

Ising density wave order

( ) /U E w= −

( ) /U E w= −

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. The AC stark shift of the atomic hyperfine levels would differ between adja-cent sites. The relative strengths of the split hyperfine absorption lines would then be a measure of the Ising order parameter.