Outline of these lectures

Ultracold fermions in optical lattices

t

U

t

Fermionic atoms in optical lattices

Experiments with fermions in optical lattice, Kohl et al., PRL 2005

Antiferromagnetic and superconducting Tc of the order of 100 K

Atoms in optical lattice

Antiferromagnetism and pairing at sub-micro Kelvin temperatures

Same microscopic model

Quantum simulations with ultracold atoms

Positive U Hubbard model

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

Antiferromagnetic insulator

D-wave superconductor

-

Fermionic Hubbard modelPhenomena predicted

Superexchange and antiferromagnetism (P.W. Anderson)

Itinerant ferromagnetism. Stoner instability (J. Hubbard)

Incommensurate spin order. Stripes (Schulz, Zaannen, Emery, Kivelson, Fradkin, White, Scalapino, Sachdev, …)

Mott state without spin order. Dynamical Mean Field Theory(Kotliar, Georges, Metzner, Vollhadt, Rozenberg, …)

d-wave pairing(Scalapino, Pines,…)

d-density wave (Affleck, Marston, Chakravarty, Laughlin,…)

Superexchange and antiferromagnetismin the Hubbard model. Large U limit

Singlet state allows virtual tunneling and regains some kinetic energy

Triplet state: virtual tunneling forbidden by Pauli principle

Effective Hamiltonian: Heisenberg model

Hubbard model for small U. Antiferromagnetic instability at half filling

Q=(p,p)

Fermi surface for n=1 Analysis of spin instabilities.Random Phase Approximation

Nesting of the Fermi surface leads to singularity

BCS-type instability for weak interaction

Hubbard model at half filling

U

TN

paramagneticMott phase

Paramagnetic Mott phase:

one fermion per sitecharge fluctuations suppressedno spin order

BCS-typetheory applies

Heisenbergmodel applies

SU(N) Magnetism withUltracold Alkaline-Earth Atoms

Alkaline-Earth atoms in optical lattice:Ex: 87Sr (I = 9/2)

|g = 1S0

|e = 3P0

698 nm150 s ~ 1 mHz

• orbital degree of freedom ⇒ spin-orbital physics→ Kugel-Khomskii model [transition metal oxides with perovskite structure]→ SU(N) Kondo lattice model [for N=2, colossal magnetoresistance in manganese oxides and heavy fermion materials]

Nuclear spin decoupled from electrons SU(N=2I+1) symmetry→ SU(N) spin models valence-bond-solid & spin-liquid phases⇒

A. Gorshkov, et al., Nature Physics 2010

Doped Hubbard model

Attraction between holes in the Hubbard model

Loss of superexchange energy from 8 bonds

Loss of superexchange energy from 7 bonds

Single plaquette: binding energy

Pairing of holes in the Hubbard model

Non-local pairing of holes

Leading istability:d-waveScalapino et al, PRB (1986)

k’

k

-k’

-kspinfluctuation

Pairing of holes in the Hubbard model

Q

BCS equation for pairing amplitude

k’

k

-k’

-kspinfluctuation

Systems close to AF instability:

c(Q) is large and positive

Dk should change sign for k’=k+Q

++-

-

dx2-y2

Stripe phases in the Hubbard model

Stripes:Antiferromagnetic domainsseparated by hole rich regions

Antiphase AF domainsstabilized by stripe fluctuations

First evidence: Hartree-Fock calculations. Schulz, Zaannen (1989)

Numerical evidence for ladders: Scalapino, White (2003); For a recent review see Fradkin arXiv:1004.1104

Possible Phase Diagram

doping

T

AF

D-SCSDW

pseudogap

n=1

After several decades we do not yet know the phase diagram

AF – antiferromagneticSDW- Spin Density Wave(Incommens. Spin Order, Stripes)D-SC – d-wave paired

Antiferromagnetic and superconducting Tc of the order of 100 K

Atoms in optical lattice

Antiferromagnetism and pairing at sub-micro Kelvin temperatures

Same microscopic model

Quantum simulations with ultracold atoms

How to detect fermion pairing

Quantum noise analysis of TOF images is more than HBT interference

Second order interference from the BCS superfluid

)'()()',( rrrr nnn

n(r)

n(r’)

n(k)

k

0),( BCSn rr

BCS

BEC

kF

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

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

Current experiments

Fermions in optical lattice. Next challenge: antiferromagnetic state

TN

U

Mott

currentexperiments

Signatures of incompressible Mott state of fermions in optical lattice

Suppression of double occupancies R. Joerdens et al., Nature (2008)

Compressibility measurementsU. Schneider et al., Science (2008)

Fermions in optical lattice. Next challenge: antiferromagnetic state

TN

U

Mott

currentexperiments

t

|U|

t

Negative U Hubbard model

Phase diagram: Micnas et al., Rev. Mod. Phys.,1990

20.04.23 27

Hubbard model with attraction: anomalous expansion

Constant Entropy

Constant Particle Number

Constant Confinement

Hackermuller et al., Science 2010

20.04.23 28

Anomalous expansion: Competition of energy and entropy

Lattice modulation experiments with fermions in optical lattice.

Related theory work: Kollath et al., PRL (2006) Huber, Ruegg, PRB (2009) Orso, Iucci, et al., PRA (2009)

Probing the Mott state of fermionsSensarma, Pekker, Lukin, Demler, PRL (2009)

Lattice modulation experimentsProbing dynamics of the Hubbard model

Measure number of doubly occupied sites

Main effect of shaking: modulation of tunneling

Modulate lattice potential

Doubly occupied sites created when frequency w matches Hubbard U

Lattice modulation experimentsProbing dynamics of the Hubbard model

R. Joerdens et al., Nature 455:204 (2008)

Mott state

Regime of strong interactions U>>t.

Mott gap for the charge forms at

Antiferromagnetic ordering at

“High” temperature regime

“Low” temperature regime

All spin configurations are equally likely.Can neglect spin dynamics.

Spins are antiferromagnetically ordered or have strong correlations

d

h Assume independent propagation of hole and doublon (neglect vertex corrections)

= +

Self-consistent Born approximation Schmitt-Rink et al (1988), Kane et al. (1989)

Spectral function for hole or doublonSharp coherent part:

dispersion set by t2/U, weight by t/U

Incoherent part:dispersion

Propagation of holes and doublons strongly affected by interaction with spin waves

Schwinger bosons Bose condensed

“Low” Temperature

Oscillations reflect shake-off processes of spin waves

“Low” Temperature

Rate of doublon production

• Sharp absorption edge due to coherent quasiparticles

• Broad continuum due to incoherent part

• Spin wave shake-off peaks

Retraceable Path Approximation Brinkmann & Rice, 1970

Consider the paths with no closed loops

Spectral Fn. of single hole

Original Experiment: R. Joerdens et al., Nature 455:204 (2008)

Theory: Sensarma et al.,PRL 103, 035303 (2009)

“High” Temperature

d

h

Reduced probability to find a singlet on neighboring sites

Radius Radius

Den

sity

Psi

ng

let

D. Pekker et al., upublished

Temperature dependenceTemperature dependence

Fermions in optical lattice.Decay of repulsively bound pairs

Ref: N. Strohmaier et al., PRL 2010

Fermions in optical lattice.Decay of repulsively bound pairs

Doublons – repulsively bound pairsWhat is their lifetime?

Excess energy U should be converted to kinetic energy of single atoms

Decay of doublon into a pair of quasiparticles requires creation of many particle-hole pairs

Direct decay isnot allowed byenergy conservation

Fermions in optical lattice.Decay of repulsively bound pairs

Experiments: N. Strohmaier et. al.

Energy carried by

spin excitations ~ J =4t2/U

Relaxation requires creation of ~U2/t2

spin excitations

Relaxation of doublon- hole pairs in the Mott state

Energy U needs to be absorbed by spin excitations

Relaxation rate

Very slow, not relevant for ETH experiments

Doublon decay in a compressible state

Excess energy U isconverted to kineticenergy of single atoms

Compressible state: Fermi liquid description

Doublon can decay into apair of quasiparticles with many particle-hole pairs

Up-p

p-h

p-h

p-h

Doublon decay in a compressible state

Perturbation theory to order n=U/6tDecay probability

Doublon Propagator Interacting “Single” Particles

Doublon decay in a compressible state

To calculate the rate: consider processes which maximize the number of particle-hole excitations

Expt: ETHZTheory: Harvard

N. Strohmaier et al., PRL 2010

Why understanding doublon decay rate is important

Prototype of decay processes with emission of many interacting particles. Example: resonance in nuclear physics: (i.e. delta-isobar)

Analogy to pump and probe experiments in condensed matter systems

Response functions of strongly correlated systems at high frequencies. Important for numerical analysis.

Important for adiabatic preparation of strongly correlated systems in optical lattices

Surprises of dynamicsin the Hubbard model

New dynamical symmetry:identical slowdown of expansionfor attractive and repulsiveinteractions

Expansion of interacting fermions in optical latticeU. Schneider et al., arXiv:1005.3545

Competition between pairing and ferromagnetic instabilities in ultracold Fermi gases near Feshbach resonances

arXiv:1005.2366

D. Pekker, M. Babadi, R. Sensarma, N. Zinner, L. Pollet, M. Zwierlein, E. Demler

Stoner model of ferromagnetismSpontaneous spin polarizationdecreases interaction energybut increases kinetic energy ofelectrons

Mean-field criterion

U N(0) = 1

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

Theoretical proposals for observing Stoner instabilitywith ultracold Fermi gases:Salasnich et. al. (2000); Sogo, Yabu (2002); Duine, MacDonald (2005); Conduit, Simons (2009); LeBlanck et al. (2009); …

Existence of Stoner type ferromagnetism in a single band model is still a subject of debate

Experiments weredone dynamically.What are implicationsof dynamics?Why spin domains could not be observed?

Is it sufficient to consider effective model with repulsive interactions when analyzing experiments?

Feshbach physics beyond effective repulsive interaction

Feshbach resonanceInteractions between atoms are intrinsically attractiveEffective repulsion appears due to low energy bound states

Example:

scattering lengthV(x)

V0 tunable by the magnetic fieldCan tune through bound state

Feshbach resonanceTwo particle bound state formed in vacuum

BCS instabilityStoner instability

Molecule formationand condensation

This talk: Prepare Fermi state of weakly interacting atoms. Quench to the BEC side of Feshbach resonance. System unstable to both molecule formation and Stoner ferromagnetism. Which instability dominates ?

Many-body instabilitiesImaginary frequencies of collective modes

Magnetic Stoner instability

Pairing instability

= + + + …

Many body instabilities near Feshbach resonance: naïve picture

Pairing (BCS) Stoner (BEC)

EF=Pairing (BCS) Stoner (BEC)

Pairing instability regularized bubble isUV divergent

To keep answers finite, we must tune together:upper momentum cut-off interaction strength U

Instability to pairing even on the BEC side

Change from bare interaction to the scattering length

Pairing instabilityIntuition: two body collisions do not lead to molecule formation on the BEC side of Feshbach resonance.Energy and momentum conservation laws can notbe satisfied.

This argument applies in vacuum. Fermi sea preventsformation of real Feshbach molecules by Pauli blocking.

Molecule Fermi sea

Stoner instability

Divergence in the scattering amplitude arises from bound state formation. Bound state is strongly affected by the Fermi sea.

Stoner instability is determined by two particlescattering amplitude

= + + + …= + + + …

Stoner instabilityRPA spin susceptibility

Interaction = Cooperon

Stoner instability

Pairing instability always dominates over pairing

If ferromagnetic domains form, they form at large q

Pairing instability vs experiments

Summary• Introduction. Magnetic and optical trapping of ultracold

atoms.• Cold atoms in optical lattices. • Bose Hubbard model. Equilibrium and dynamics• Bose mixtures in optical lattices Quantum magnetism of ultracold atoms.• Detection of many-body phases using noise correlations• Experiments with low dimensional systems Interference experiments. Analysis of high order correlations• Fermions in optical lattices Magnetism and pairing in systems with repulsive interactions.

Current experiments: paramgnetic Mott state, nonequilibrium dynamics.

• Dynamics near Fesbach resonance. Competition of Stoner instability and pairing

• Detection of many-body phases• Nonequilibrium dynamics

Emphasis of these lectures: