Outline of these lectures

• Fermions in optical lattices Magnetism Pairing in systems with repulsive interactions Current experiments: Paramagnetic Mott state • Experiments on nonequilibrium fermion dynamics Lattice modulation experiments Doublon decay • Stoner instability

Ultracold fermions in optical lattices

Fermionic atoms in optical lattices

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

Quantum simulations with ultracold atoms Atoms in optical lattice Antiferromagnetic and superconducting Tc of the order of 100 K Antiferromagnetism and pairing at sub-micro Kelvin temperatures

Same microscopic model

Positive U Hubbard model

Possible phase diagram. Scalapino, Phys. Rep. 250:329 (1995) Antiferromagnetic insulator D-wave superconductor

-

Fermionic Hubbard model Phenomena 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 antiferromagnetism in 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 Fermi surface for n=1 Analysis of spin instabilities.

Random Phase Approximation Q=(p,p) Nesting of the Fermi surface leads to singularity BCS-type instability for weak interaction

Hubbard model at half filling

T N BCS-type theory applies paramagnetic Mott phase Paramagnetic Mott phase: one fermion per site charge fluctuations suppressed no spin order U Heisenberg model applies

SU(N) Magnetism with Ultracold Alkaline-Earth Atoms A. Gorshkov, et al., Nature Physics 2010 Ex: 87 Sr (I = 9/2) |e > = 3 P 0 Alkaline-Earth atoms in optical lattice: 698 nm 150 s ~ 1 mHz |g > = 1 S 0 Nuclear spin decoupled from electrons SU(N=2I+1) symmetry → SU(N) spin models ⇒ valence-bond-solid & spin-liquid phases • 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]

Doped Hubbard model

Attraction between holes in the Hubbard model Loss of superexchange energy from 8 bonds Single plaquette: binding energy Loss of superexchange energy from 7 bonds

Pairing of holes in the Hubbard model Leading istability: d-wave Scalapino et al, PRB (1986) Non-local pairing of holes k’ k spin fluctuation k’ -k

+ Q Pairing of holes in the Hubbard model BCS equation for pairing amplitude d x2-y2 + k’ k’ k spin fluctuation -k Systems close to AF instability: c (Q) is large and positive D k should change sign for k’=k+Q

Stripe phases in the Hubbard model Stripes: Antiferromagnetic domains separated by hole rich regions Antiphase AF domains stabilized 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 T AF – antiferromagnetic SDW- Spin Density Wave (Incommens. Spin Order, Stripes) D-SC – d-wave paired AF pseudogap SDW D-SC n=1 doping After several decades we do not yet know the phase diagram

Quantum simulations with ultracold atoms Atoms in optical lattice Antiferromagnetic and superconducting Tc of the order of 100 K Antiferromagnetism and pairing at sub-micro Kelvin temperatures

Same microscopic model

How to detect fermion pairing

Quantum noise analysis of TOF images is more than HBT interference

Second order interference from the BCS superfluid Theory: Altman et al., PRA 70:13603 (2004)

n(k) n(r’) k F BCS BEC k n(r) r

n

, )

Momentum correlations in paired fermions

Greiner et al., PRL 94:110401 (2005)

Fermion pairing in an optical lattice

Second Order Interference In 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

T N Mott current experiments U

Signatures of incompressible Mott state of fermions in optical lattice Suppression of double occupancies R. Joerdens et al., Nature (2008) Compressibility measurements U. Schneider et al., Science (2008)

Fermions in optical lattice. Next challenge: antiferromagnetic state

T N Mott current experiments U

Negative U Hubbard model

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

Hubbard model with attraction: anomalous expansion

Hackermuller et al., Science 2010 Constant Entropy Constant Particle Number Constant Confinement 26.04.2020

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Anomalous expansion: Competition of energy and entropy

26.04.2020

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Lattice modulation experiments with fermions in optical lattice.

Probing the Mott state of fermions Sensarma, Pekker, Lukin, Demler, PRL (2009) Related theory work: Kollath et al., PRL (2006) Huber, Ruegg, PRB (2009) Orso, Iucci, et al., PRA (2009)

Lattice modulation experiments Probing dynamics of the Hubbard model Modulate lattice potential Measure number of doubly occupied sites Main effect of shaking: modulation of tunneling Doubly occupied sites created when frequency w matches Hubbard

U

Lattice modulation experiments Probing 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 All spin configurations are equally likely.

Can neglect spin dynamics.

“Low” temperature regime Spins are antiferromagnetically ordered or have strong correlations

d

“Low” Temperature Schwinger bosons Bose condensed Propagation of holes and doublons strongly affected by interaction with spin waves

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 doublon Sharp coherent part: dispersion set by

t 2 /U

, weight by

t/U

Incoherent part: dispersion 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

d h

“High” Temperature

Retraceable Path Approximation

Brinkmann & Rice, 1970

Consider the paths with no closed loops Original Experiment: R. Joerdens et al., Nature 455:204 (2008) Theory: Sensarma et al., PRL 103, 035303 (2009) Spectral Fn. of single hole

Temperature dependence Reduced probability to find a singlet on neighboring sites Radius Radius D. Pekker et al., upublished

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 pairs What is their lifetime?

Direct decay is not allowed by energy conservation 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

Fermions in optical lattice.

Decay of repulsively bound pairs

Experiments: N. Strohmaier et. al.

Relaxation of doublon- hole pairs in the Mott state Energy U needs to be absorbed by spin excitations  Energy carried by spin excitations ~ J =4t 2 /U Relaxation rate  Relaxation requires creation of ~U 2 spin excitations /t 2 Very slow, not relevant for ETH experiments

Doublon decay in a compressible state Excess energy U is converted to kinetic energy of single atoms Compressible state: Fermi liquid description Doublon can decay into a pair of quasiparticles with many particle-hole pairs

U p h p h p h p p

Doublon decay in a compressible state Perturbation theory to order n=U/6t Decay probability

Doublon Propagator Interacting “Single” Particles

Doublon decay in a compressible state N. Strohmaier et al., PRL 2010 Expt: ETHZ Theory: Harvard To calculate the rate: consider processes which maximize the number of particle-hole excitations

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 dynamics in the Hubbard model

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

New dynamical symmetry: identical slowdown of expansion for attractive and repulsive interactions

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 ferromagnetism

Spontaneous spin polarization decreases interaction energy but increases kinetic energy of electrons Mean-field criterion

U N(0) = 1

U – interaction strength N(0) – density of states at Fermi level Existence of Stoner type ferromagnetism in a single band model is still a subject of debate Theoretical proposals for observing Stoner instability with ultracold Fermi gases: Salasnich et. al. (2000); Sogo, Yabu (2002); Duine, MacDonald (2005); Conduit, Simons (2009); LeBlanck et al. (2009); …

Experiments were done dynamically.

What are implications of 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 resonance

Interactions between atoms are intrinsically attractive Effective repulsion appears due to low energy bound states Example:

V(x)

scattering length

V 0

tunable by the magnetic field Can tune through bound state

Feshbach resonance

Two particle bound state formed in vacuum Stoner instability BCS instability Molecule formation and 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 instabilities

Imaginary frequencies of collective modes Magnetic Stoner instability Pairing instability = + + + …

Many body instabilities near Feshbach resonance: naïve picture Pairing (BCS) Stoner (BEC) E F = Pairing (BCS) Stoner (BEC)

Pairing instability regularized

bubble is UV divergent To keep answers finite, we must tune together: upper momentum cut-off interaction strength U Change from bare interaction to the scattering length Instability to pairing even on the BEC side

Pairing instability

Intuition: two body collisions do not lead to molecule formation on the BEC side of Feshbach resonance.

Energy and momentum conservation laws can not be satisfied.

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

Molecule Fermi sea

Stoner instability

Stoner instability is determined by two particle scattering amplitude Divergence in the scattering amplitude arises from bound state formation. Bound state is strongly affected by the Fermi sea.

Stoner instability

RPA 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 Emphasis of these lectures : • Detection of many-body phases • Nonequilibrium dynamics