Eugene Demler Harvard University

Main collaborators:

Anatoli Polkovnikov Harvard/Boston UniversityEhud Altman Harvard/WeizmannDaw-Wei Wang Harvard/Tsing-Hua UniversityVladimir Gritsev HarvardAdilet Imambekov HarvardRyan Barnett Harvard/CaltechMikhail Lukin Harvard

Simulations of strongly correlated electron systems using cold atoms

Strongly correlated electron systems

“Conventional” solid state materials

Bloch theorem for non-interacting electrons in a periodic potential

B

VH

I

d

First semiconductor transistor

EF

Metals

EF

Insulators andSemiconductors

Consequences of the Bloch theorem

“Conventional” solid state materialsElectron-phonon and electron-electron interactions are irrelevant at low temperatures

kx

ky

kF

Landau Fermi liquid theory: when frequency and temperature are smaller than EF electron systems

are equivalent to systems of non-interacting fermions

Ag Ag

Ag

Non Fermi liquid behavior in novel quantum materials

CeCu2Si2. Steglich et al.,

Z. Phys. B 103:235 (1997)

UCu3.5Pd1.5

Andraka, Stewart, PRB 47:3208 (93)

Violation of the Wiedemann-Franz lawin high Tc superconductorsHill et al., Nature 414:711 (2001)

Puzzles of high temperature superconductors

Maple, JMMM 177:18 (1998)Unusual “normal” state

Resistivity, opical conductivity,Lack of sharply defined quasiparticles,Nernst effect

Mechanism of Superconductivity

High transition temperature,retardation effect, isotope effect,role of elecron-electron and electron-phonon interactions

Competing orders

Role of magnetsim, stripes,possible fractionalization

Applications of quantum materials:High Tc superconductors

Applications of quantum materials: Ferroelectric RAM

Non-Volatile Memory

High Speed Processing

FeRAM in Smart Cards

V+ + + + + + + +

_ _ _ _ _ _ _ _

Modeling strongly correlated systems using cold atoms

Bose-Einstein condensation

Cornell et al., Science 269, 198 (1995)

Ultralow density condensed matter system

Interactions are weak and can be described theoretically from first principles

New Era in Cold Atoms ResearchFocus on Systems with Strong Interactions

• Atoms in optical lattices

• Feshbach resonances

• Low dimensional systems

• Systems with long range dipolar interactions

• Rotating systems

Feshbach resonance and fermionic condensates Greiner et al., Nature 426:537 (2003); Ketterle et al., PRL 91:250401 (2003)

Ketterle et al.,Nature 435, 1047-1051 (2005)

One dimensional systems

Strongly interacting regime can be reached for low densities

One dimensional systems in microtraps.Thywissen et al., Eur. J. Phys. D. (99);Hansel et al., Nature (01);Folman et al., Adv. At. Mol. Opt. Phys. (02)

1D confinement in optical potentialWeiss et al., Science (05);Bloch et al., Esslinger et al.,

Atoms in optical lattices

Theory: Jaksch et al. PRL (1998)

Experiment: Kasevich et al., Science (2001); Greiner et al., Nature (2001); Phillips et al., J. Physics B (2002) Esslinger et al., PRL (2004); and many more …

Strongly correlated systemsAtoms in optical latticesElectrons in Solids

Simple metalsPerturbation theory in Coulomb interaction applies. Band structure methods wotk

Strongly Correlated Electron SystemsBand structure methods fail.

Novel phenomena in strongly correlated electron systems:

Quantum magnetism, phase separation, unconventional superconductivity,high temperature superconductivity, fractionalization of electrons …

New Era in Cold Atoms ResearchFocus on Systems with Strong Interactions

Goals

• Resolve long standing questions in condensed matter physics (e.g. origin of high temperature superconductivity)

• Resolve matter of principle questions (e.g. existence of spin liquids in two and three dimensions)

• Study new phenomena in strongly correlated systems (e.g. coherent far from equilibrium dynamics)

Outline• Introduction. Cold atoms in optical lattices. Bose Hubbard

model• Two component Bose mixtures

Quantum magnetism. Competing orders. Fractionalized phases

• Fermions in optical lattices

Pairing in systems with repulsive interactions. High Tc mechanism • Boson-Fermion mixtures

Polarons. Competing orders

• Interference experiments with fluctuating BEC

Analysis of correlations beyond mean-field

• Moving condensates in optical lattices

Non equilibrium dynamics of interacting many-body systems

Emphasis: detection and characterzation of many-body states

Atoms in optical lattices. Bose Hubbard model

Bose Hubbard model

tunneling of atoms between neighboring wells

repulsion of atoms sitting in the same well

U

t

4

Bose Hubbard model. Mean-field phase diagram

1n

U

02

0

M.P.A. Fisher et al.,PRB40:546 (1989)

MottN=1

N=2

N=3

Superfluid

Superfluid phase

Mott insulator phase

Weak interactions

Strong interactions

Mott

Mott

Set .

Bose Hubbard model

Hamiltonian eigenstates are Fock states

U

2 4

Bose Hubbard Model. Mean-field phase diagram

Particle-hole excitation

Mott insulator phase

41n

U

2

0

MottN=1

N=2

N=3

Superfluid

Mott

Mott

Tips of the Mott lobes

Gutzwiller variational wavefunction

Normalization

Interaction energy

Kinetic energy

z – number of nearest neighbors

Phase diagram of the 1D Bose Hubbard model. Quantum Monte-Carlo study

Batrouni and Scaletter, PRB 46:9051 (1992)

Optical lattice and parabolic potential

41n

U

2

0

N=1

N=2

N=3

SF

MI

MI

Jaksch et al., PRL 81:3108 (1998)

Superfluid to Insulator transitionGreiner et al., Nature 415:39 (2002)

U

1n

t/U

SuperfluidMott insulator

Time of flight experiments

Quantum noise interferometry of atoms in an optical lattice

Second order coherence

Second order coherence in the insulating state of bosons.Hanburry-Brown-Twiss experiment

Theory: Altman et al., PRA 70:13603 (2004)

Experiment: Folling et al., Nature 434:481 (2005)

Hanburry-Brown-Twiss stellar interferometer

Hanburry-Brown-Twiss interferometer

Second order coherence in the insulating state of bosons

Bosons at quasimomentum expand as plane waves

with wavevectors

First order coherence:

Oscillations in density disappear after summing over

Second order coherence:

Correlation function acquires oscillations at reciprocal lattice vectors

Second order coherence in the insulating state of bosons.Hanburry-Brown-Twiss experiment

Theory: Altman et al., PRA 70:13603 (2004)

Experiment: Folling et al., Nature 434:481 (2005)

Effect of parabolic potential on the second order coherence

Experiment: Spielman, Porto, et al.,Theory: Scarola, Das Sarma, Demler, cond-mat/0602319

Width of the correlation peak changes across the transition, reflecting the evolution of Mott domains

Width of the noise peaks

0 200 400 600 800 1000 1200

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

Interference of an array of independent condensates

Hadzibabic et al., PRL 93:180403 (2004)

Smooth structure is a result of finite experimental resolution (filtering)

0 200 400 600 800 1000 1200-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Extended Hubbard Model

- on site repulsion - nearest neighbor repulsion

Checkerboard phase:

Crystal phase of bosons. Breaks translational symmetry

Extended Hubbard model. Mean field phase diagram

van Otterlo et al., PRB 52:16176 (1995)

Hard core bosons.

Supersolid – superfluid phase with broken translational symmetry

Extended Hubbard model. Quantum Monte Carlo study

Sengupta et al., PRL 94:207202 (2005)Hebert et al., PRB 65:14513 (2002)

Dipolar bosons in optical lattices

Goral et al., PRL88:170406 (2002)

How to detect a checkerboard phase

Correlation Function Measurements

Magnetism in condensed matter systems

Ferromagnetism

Magnetic memory in hard drives.Storage density of hundreds of billions bits per square inch.

Magnetic needle in a compass

Stoner model of ferromagnetism

Spontaneous spin polarizationdecreases interaction energybut increases kinetic energy ofelectrons

Mean-field criterion I N(0) = 1

I – interaction strengthN(0) – density of states at the Fermi level

Antiferromagnetism

High temperature superconductivity in cuprates is always foundnear an antiferromagnetic insulating state

Maple, JMMM 177:18 (1998)

( + )

Antiferromagnetism

Antiferromagnetic Heisenberg model

( - )S =

( + )t =

AF =

AF = S t

Antiferromagnetic state breaks spin symmetry. It does not have a well defined spin

Spin liquid states

Alternative to classical antiferromagnetic state: spin liquid states

Properties of spin liquid states:

• fractionalized excitations• topological order• gauge theory description

Systems with geometric frustration

?

Spin liquid behavior in systems with geometric frustration

Kagome lattice

SrCr9-xGa3+xO19

Ramirez et al. PRL (90)Broholm et al. PRL (90)Uemura et al. PRL (94)

ZnCr2O4

A2Ti2O7

Ramirez et al. PRL (02)

Pyrochlore lattice

Engineering magnetic systems using cold atoms in an optical lattice

Spin interactions using controlled collisions

Experiment: Mandel et al., Nature 425:937(2003)

Theory: Jaksch et al., PRL 82:1975 (1999)

Effective spin interaction from the orbital motion.Cold atoms in Kagome lattices

Santos et al., PRL 93:30601 (2004)

Damski et al., PRL 95:60403 (2005)

t

t

Two component Bose mixture in optical latticeExample: . Mandel et al., Nature 425:937 (2003)

Two component Bose Hubbard model

Quantum magnetism of bosons in optical lattices

Duan et al., PRL (2003)

• Ferromagnetic• Antiferromagnetic

Kuklov and Svistunov, PRL (2003)

Exchange Interactions in Solids

antibonding

bonding

Kinetic energy dominates: antiferromagnetic state

Coulomb energy dominates: ferromagnetic state

Two component Bose mixture in optical lattice.Mean field theory + Quantum fluctuations

2 nd order line

Hysteresis

1st order

Altman et al., NJP 5:113 (2003)

Coherent spin dynamics in optical lattices

Widera et al., cond-mat/0505492

atoms in the F=2 state

How to observe antiferromagnetic order of cold atoms in an optical lattice?

Second order coherence in the insulating state of bosons.Hanburry-Brown-Twiss experiment

Theory: Altman et al., PRA 70:13603 (2004) See also Bach, Rzazewski, PRL 92:200401 (2004)

Experiment: Folling et al., Nature 434:481 (2005)

See also Hadzibabic et al., PRL 93:180403 (2004)

Probing spin order of bosons

Correlation Function Measurements

Engineering exotic phases

• Optical lattice in 2 or 3 dimensions: polarizations & frequenciesof standing waves can be different for different directions

ZZ

YY

• Example: exactly solvable modelKitaev (2002), honeycomb lattice with

H Jx

i, jx

ix j

x Jy

i, jy

iy j

y Jz

i, jz

iz j

z

• Can be created with 3 sets of standing wave light beams !• Non-trivial topological order, “spin liquid” + non-abelian anyons …those has not been seen in controlled experiments

Fermionic atoms in optical lattices

Pairing in systems with repulsive interactions. Unconventional pairing. High Tc mechanism

Fermionic atoms in a three dimensional optical lattice

Kohl et al., PRL 94:80403 (2005)

Fermions with attractive interaction

U

tt

Hofstetter et al., PRL 89:220407 (2002)

Highest transition temperature for

Compare to the exponential suppresion of Tc w/o a lattice

Reaching BCS superfluidity in a lattice

6Li

40K

Li in CO2 lattice

K in NdYAG lattice

Turning on the lattice reduces the effective atomic temperature

Superfluidity can be achived even with a modest scattering length

Fermions with repulsive interactions

t

U

tPossible d-wave pairing of fermions

Picture courtesy of UBC Superconductivity group

High temperature superconductors

Superconducting Tc 93 K

Hubbard model – minimal model for cuprate superconductors

P.W. Anderson, cond-mat/0201429

After many years of work we still do not understand the fermionic Hubbard model

Positive U Hubbard model

Possible phase diagram. Scalapino, Phys. Rep. 250:329 (1995)

Antiferromagnetic insulator

D-wave superconductor

Second order correlations in the BCS superfluid

)'()()',( rrrr nnn

n(r)

n(r’)

n(k)

k

0),( BCSn rr

BCS

BEC

kF

Momentum correlations in paired fermionsGreiner et al., PRL 94:110401 (2005)

Fermion pairing in an optical lattice

Second Order InterferenceIn the TOF images

Normal State

Superfluid State

measures the Cooper pair wavefunction

One can identify unconventional pairing

Boson Fermion mixtures

Fermions interacting with phonons.Polarons. Competing orders

Boson Fermion mixtures

BEC

Experiments: ENS, Florence, JILA, MIT, Rice, …

Bosons provide cooling for fermionsand mediate interactions. They createnon-local attraction between fermions

Charge Density Wave Phase

Periodic arrangement of atoms

Non-local Fermion Pairing

P-wave, D-wave, …

Boson Fermion mixtures

“Phonons” :Bogoliubov (phase) mode

Effective fermion-”phonon” interaction

Fermion-”phonon” vertex Similar to electron-phonon systems

Boson Fermion mixtures in 1d optical latticesCazalila et al., PRL (2003); Mathey et al., PRL (2004)

Spinless fermions Spin ½ fermions

Note: Luttinger parameters can be determined using correlation functionmeasurements in the time of flight experiments. Altman et al. (2005)

BF mixtures in 2d optical lattices

40K -- 87Rb 40K -- 23Na

=1060 nm(a) =1060nm

(b) =765.5nm

Wang, Lukin, Demler, PRA (1972)

Systems of cold atoms with strong interactions and correlations

Goals

Resolve long standing questions in condensed matter physics (e.g. origin of high temperature superconductivity)

Resolve matter of principle questions (e.g. existence of spin liquids in two and three dimensions)

Study new phenomena in strongly correlated systems• Interference experiments with fluctuating BEC Analysis of high order correlation functions in low dimensional systems• Moving condensates in optical lattices Non equilibrium dynamics of interacting many-body systems

Analysis of high order correlation functions in low dimensional systems

Interference experiments with fluctuating BEC

Interference of two independent condensates

Andrews et al., Science 275:637 (1997)

Interference of two independent condensates

1

2

r

r+d

d

r’

Clouds 1 and 2 do not have a well defined phase difference.However each individual measurement shows an interference pattern

Amplitude of interference fringes, , contains information about phase fluctuations within individual condensates

y

x

Interference of one dimensional condensates

x1

d Experiments: Schmiedmayer et al., Nature Physics (2005)

x2

Interference amplitude and correlations

For identical condensates

Instantaneous correlation function

L

Polkovnikov, Altman, Demler, PNAS (2006)

For impenetrable bosons and

Interference between Luttinger liquidsLuttinger liquid at T=0

K – Luttinger parameter

Luttinger liquid at finite temperature

For non-interacting bosons and

Analysis of can be used for thermometry

L

Luttinger parameter K may beextracted from the angular dependence of

Rotated probe beam experiment

For large imaging angle, ,

Interference between two-dimensional BECs at finite temperature.Kosteritz-Thouless transition

Interference of two dimensional condensates

Ly

Lx

Lx

Experiments: Stock, Hadzibabic, Dalibard, et al., cond-mat/0506559

Probe beam parallel to the plane of the condensates

Gati, Oberthaler, et al., cond-mat/0601392

Interference of two dimensional condensates.Quasi long range order and the KT transition

Ly

Lx

Below KT transitionAbove KT transition

Theory: Polkovnikov, Altman, Demler, PNAS (2006)

x

z

Time of

flight

low temperature higher temperature

Typical interference patterns

Experiments with 2D Bose gasHaddzibabic et al., Nature (2006)

integration

over x axis

Dx

z

z

integration

over x axisz

x

integration distance Dx

(pixels)

Contrast afterintegration

0.4

0.2

00 10 20 30

middle Tlow T

high T

integration

over x axis z

Experiments with 2D Bose gasHadzibabic et al., Nature (2006)

fit by:

integration distance Dx

Inte

grat

ed c

ontr

ast 0.4

0.2

00 10 20 30

low Tmiddle T

high T

if g1(r) decays exponentially with :

if g1(r) decays algebraically or exponentially with a large :

Exponent

central contrast

0.5

0 0.1 0.2 0.3

0.4

0.3high T low T

2

21

2 1~),0(

1~

x

D

x Ddxxg

DC

x

“Sudden” jump!?

Experiments with 2D Bose gasHadzibabic et al., Nature (2006)

Exponent

central contrast

0.5

0 0.1 0.2 0.3

0.4

0.3 high T low T

He experiments:universal jump in

the superfluid density

T (K)1.0 1.1 1.2

1.0

0

c.f. Bishop and Reppy

Experiments with 2D Bose gasHadzibabic et al., Nature (2006)

Ultracold atoms experiments:jump in the correlation function.

KT theory predicts =1/4 just below the transition

Experiments with 2D Bose gas. Proliferation of thermal vortices Haddzibabic et al., Nature (2006)

Fraction of images showing at least one dislocation

Exponent

0.5

0 0.1 0.2 0.3

0.4

0.3

central contrast

The onset of proliferation coincides with shifting to 0.5!

0

10%

20%

30%

central contrast

0 0.1 0.2 0.3 0.4

high T low T

Z. Hadzibabic et al., Nature (2006)

Interference between two interacting one dimensional Bose liquids

Full distribution function of the interference amplitude

Gritsev, Altman, Demler, Polkovnikov, cond-mat/0602475

Higher moments of interference amplitude

L Higher moments

Changing to periodic boundary conditions (long condensates)

Explicit expressions for are available but cumbersome Fendley, Lesage, Saleur, J. Stat. Phys. 79:799 (1995)

is a quantum operator. The measured value of will fluctuate from shot to shot.Can we predict the distribution function of ?

Impurity in a Luttinger liquid

Expansion of the partition function in powers of g

Partition function of the impurity contains correlation functions taken at the same point and at different times. Momentsof interference experiments come from correlations functionstaken at the same time but in different points. Euclidean invarianceensures that the two are the same

Relation between quantum impurity problemand interference of fluctuating condensates

Distribution function of fringe amplitudes

Distribution function can be reconstructed fromusing completeness relations for the Bessel functions

Normalized amplitude of interference fringes

Relation to the impurity partition function

is related to the single particle Schroedinger equation Dorey, Tateo, J.Phys. A. Math. Gen. 32:L419 (1999) Bazhanov, Lukyanov, Zamolodchikov, J. Stat. Phys. 102:567 (2001)

Spectral determinant

Bethe ansatz solution for a quantum impurity can be obtained from the Bethe ansatz followingZamolodchikov, Phys. Lett. B 253:391 (91); Fendley, et al., J. Stat. Phys. 79:799 (95)Making analytic continuation is possible but cumbersome

Interference amplitude and spectral determinant

0 1 2 3 4

P

roba

bilit

y P

(x)

x

K=1 K=1.5 K=3 K=5

Evolution of the distribution function

Narrow distributionfor .Approaches Gumbledistribution. Width

Wide Poissoniandistribution for

When K>1, is related to Q operators of CFT with c<0. This includes 2D quantum gravity, non-intersecting loop model on 2D lattice, growth of randomfractal stochastic interface, high energy limit of multicolor QCD, …

Yang-Lee singularity

2D quantum gravity,non-intersecting loops on 2D lattice

correspond to vacuum eigenvalues of Q operators of CFT Bazhanov, Lukyanov, Zamolodchikov, Comm. Math. Phys.1996, 1997, 1999

From interference amplitudes to conformal field theories

Moving condensates in optical lattices Non equilibrium coherent dynamics

of interacting many-body systems

Atoms in optical lattices. Bose Hubbard model

Theory: Jaksch et al. PRL 81:3108(1998)

Experiment: Kasevich et al., Science (2001) Greiner et al., Nature (2001) Cataliotti et al., Science (2001) Phillips et al., J. Physics B (2002) Esslinger et al., PRL (2004), …

Equilibrium superfluid to insulator transition

1n

t/U

SuperfluidMott insulator

Theory: Fisher et al. PRB (89), Jaksch et al. PRL (98)Experiment: Greiner et al. Nature (01)

U

Moving condensate in an optical lattice. Dynamical instability

v

Theory: Niu et al. PRA (01), Smerzi et al. PRL (02)Experiment: Fallani et al. PRL (04)

Related experiments byEiermann et al, PRL (03)

This discussion: This discussion: How to connectHow to connect the the dynamical instabilitydynamical instability (irreversible, classical) (irreversible, classical)to the to the superfluid to Mott transitionsuperfluid to Mott transition (equilibrium, quantum) (equilibrium, quantum)

U/t

p

SF MI

???Possible experimental

sequence:

Unstable

???

p

U/J

Stable

SF MI

This discussion

Superconductor to Insulator transition in thin films

Marcovic et al., PRL 81:5217 (1998)

Bi films

Superconducting filmsof different thickness

d

Linear stability analysis: States with p> are unstable

Classical limit of the Hubbard model. Discreet Gross-Pitaevskii equation

Current carrying states

r

Dynamical instability

Amplification ofdensity fluctuations

unstableunstable

GP regime . Maximum of the current for .

When we include quantum fluctuations, the amplitude of the order parameter is suppressed

Dynamical instability for integer filling

decreases with increasing phase gradient

Order parameter for a current carrying state

Current

SF MI

p

U/J

Dynamical instability for integer filling

0.0 0.1 0.2 0.3 0.4 0.5

p*

I(p)

s(p)

sin(p)

Condensate momentum p/

Dynamical instability occurs for

Vicinity of the SF-I quantum phase transition. Classical description applies for

Dynamical instability. Gutzwiller approximation

0.0

0.1

0.2

0.3

0.4

0.5

0.0 0.2 0.4 0.6 0.8 1.0

d=3

d=2

d=1

unstable

stable

U/Uc

p/

Wavefunction

Time evolution

Phase diagram. Integer filling

We look for stability against small fluctuations

The first instability develops near the edges, where N=1

0 100 200 300 400 500

-0.2

-0.1

0.0

0.1

0.2

0.00 0.17 0.34 0.52 0.69 0.86

Cen

ter

of M

ass

Mom

entu

m

Time

N=1.5 N=3

U=0.01 tJ=1/4

Gutzwiller ansatz simulations (2D)

Optical lattice and parabolic trap. Gutzwiller approximation

Beyond semiclassical equations. Current decay by tunneling

pha

se

jpha

se

jpha

se

j

Current carrying states are metastable. They can decay by thermal or quantum tunneling

Thermal activation Quantum tunneling

S – classical action corresponding to the motion in an inverted potential.

Decay of current by quantum tunnelingp

hase

j

Escape from metastable state by quantum tunneling.

WKB approximation

pha

se

j

Quantumphase slip

Decay rate from a metastable state. Example

0

22 3

0

1 ( ) 0

2 c

dxS d x bx p p

m d

For d>1 we have to include transverse directions. Need to excite many chains to create a phase slip

The transverse size of the phase slip diverges near a phase slip. We can use continuum approximation to treat transverse directions

|| cos ,J J p

J J

Longitudinal stiffness is much smaller than the transverse.

SF MI

p

U/J

Weakly interacting systems. Gross-Pitaevskii regime.Decay of current by quantum tunneling

Quantum phase slips are strongly suppressed in the GP regime

Fallani et al., PRL (04)

This state becomes unstable at corresponding to the

maximum of the current:

13cp

2 2 21 .I p p p

Close to a SF-Mott transition we can use an effective relativistivc GL theory (Altman, Auerbach, 2004)

Strongly interacting regime. Vicinity of the SF-Mott transition

SF MI

p

U/J

Metastable current carrying state:2 21 ip xp e

Strong broadening of the phase transition in d=1 and d=2

is discontinuous at the transition. Phase slips are not important.Sharp phase transition

- correlation length

SF MI

p

U/J

Strongly interacting regime. Vicinity of the SF-Mott transitionDecay of current by quantum tunneling

Action of a quantum phase slip in d=1,2,3

Decay of current by quantum tunneling

0.0

0.1

0.2

0.3

0.4

0.5

0.0 0.2 0.4 0.6 0.8 1.0

d=3

d=2

d=1

unstable

stable

U/Uc

p/

Decay of current by thermal activationp

hase

j

Escape from metastable state by thermal activation

pha

se

j

Thermalphase slip

E

Thermally activated current decay. Weakly interacting regime

E

Activation energy in d=1,2,3

Thermal fluctuations lead to rapid decay of currents

Crossover from thermal to quantum tunneling

Thermalphase slip

Phys. Rev. Lett. (2004)

Decay of current by thermal fluctuations

Conclusions

We understand well: electron systems in semiconductors and simple metals.Interaction energy is smaller than the kinetic energy. Perturbation theory works

We do not understand: strongly correlated electron systems in novel materials.Interaction energy is comparable or larger than the kinetic energy.Many surprising new phenomena occur, including high temperaturesuperconductivity, magnetism, fractionalization of excitations

Our big goal is to develop a general framework for understanding stronglycorrelated systems. This will be important far beyond AMO and condensed matter

Ultracold atoms have energy scales of 10-6K, compared to 104 K forelectron systems. However, by engineering and studying strongly interactingsystems of cold atoms we should get insights into the mysterious propertiesof novel quantum materials