Outline of these lectures Introduction. Systems of ultracold atoms. Cold atoms in optical lattices. Bose Hubbard model Bose mixtures in optical lattices Detection of many-body phases using noise correlations Experiments with low dimensional systems 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 Experiments with fermions in optical lattice, Kohl et al., PRL 2005

Same microscopic model Quantum simulations with ultracold atoms 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

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

Ultracold Alkaline-Earth Atoms SU(N) Magnetism with Ultracold Alkaline-Earth Atoms A. Gorshkov, et al., Nature Physics 2010 Ex: 87Sr (I = 9/2) |g> = 1S0 |e> = 3P0 698 nm 150 s ~ 1 mHz Alkaline-Earth atoms in optical lattice: 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 Loss of superexchange energy from 7 bonds Single plaquette: binding energy

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

Pairing of holes in the Hubbard model BCS equation for pairing amplitude Q k’ k -k’ -k spin fluctuation - + + - 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 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 doping T AF D-SC SDW pseudogap n=1 AF – antiferromagnetic SDW- Spin Density Wave (Incommens. Spin Order, Stripes) D-SC – d-wave paired After several decades we do not yet know the phase diagram

Same microscopic model Quantum simulations with ultracold atoms 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

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) n(r’) k F k BCS BEC

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 TN U Mott current experiments

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 TN U Mott current experiments

Negative U Hubbard model 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 11.01.2019

Anomalous expansion: Competition of energy and entropy 11.01.2019

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

“Low” Temperature d h = + Self-consistent Born approximation Schwinger bosons Bose condensed d h Propagation of holes and doublons strongly affected by interaction with spin waves 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 t2/U, weight by t/U Incoherent part: dispersion Oscillations reflect shake-off processes of spin waves

“Low” Temperature Rate of doublon production Spin wave shake-off peaks Sharp absorption edge due to coherent quasiparticles Broad continuum due to incoherent part Spin wave shake-off peaks

“High” Temperature d h 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 Density Psinglet 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 Magnestism is a goal but we have problems 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 Relaxation requires creation of ~U2/t2 spin excitations Energy carried by spin excitations ~ J =4t2/U Relaxation rate 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 U p-p p-h Doublon can decay into a pair of quasiparticles with many particle-hole pairs

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: scattering length V(x) V0 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) EF=

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

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 Kanamori’s counter-argument: renormalization of U. then Theoretical proposals for observing Stoner instability with cold gases: Salasnich et. al. (2000); Sogo, Yabu (2002); Duine, MacDonald (2005); Conduit, Simons (2009); LeBlanck et al. (2009); … Recent work on hard sphere potentials: Pilati et al. (2010); Chang et al. (2010)

Experiments were done dynamically. What are implications of dynamics? Why spin domains could not be observed? Earlier work by C. Salomon et al., 2003

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: scattering length V(x) V0 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 ?

Pair formation

Two-particle scattering in vacuum k -k p -p Microscopic Hamiltonian Schrödinger equation

T-matrix Lippman-Schwinger equation k -k -p’ -p p p’ On-shell T-matrix. Universal low energy expression For positive scattering length bound state at appears as a pole in the T-matrix

Cooperon Two particle scattering in the presence of a Fermi sea p k -k Need to make sure that we do not include interaction effects on the Fermi liquid state in scattered state energy

Cooperon Grand canonical ensemble Define Cooperon equation

Cooperon vs T-matrix k -k -p’ -p p p’

Cooper channel response function Linear response theory Induced pairing field Response function Poles of the Cooper channel response function are given by

Cooper channel response function Linear response theory Poles of the response function, , describe collective modes Time dependent dynamics When the mode frequency has negative imaginary part, the system is unstable

Pairing instability regularized BCS side Instability rate coincides with the equilibrium gap (Abrikosov, Gorkov, Dzyaloshinski) Instability to pairing even on the BEC side Related work: Lamacraft, Marchetti, 2008

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

Pairing instability Time dependent variational wavefunction Time dependence of uk(t) and vk(t) due to DBCS(t) For small DBCS(t):

Pairing instability From wide to Effects of finite narrow resonances temperature Three body recombination as in Shlyapnikov et al., 1996; Petrov, 2003; Esry 2005

Magnetic instability

Stoner instability. Naïve theory Linear response theory Spin response function Spin collective modes are given by the poles of response function Negative imaginary frequencies correspond to magnetic instability

RPA analysis for Stoner instability Self-consistent equation on response function Spin susceptibility for non-interacting gas RPA expression for the spin response function

Quench dynamics across Stoner instability Stoner criterion For U>Uc unstable collective modes Unstable modes determine characteristic lengthscale of magnetic domains

Stoner quench dynamics in D=3 Unphysical divergence of the instability rate at unitarity

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

Relation to experiments

Pairing instability vs experiments

Schwinger bosons and Slave Fermions Constraint : Singlet Creation Boson Hopping

Schwinger bosons and slave fermions Fermion hopping Propagation of holes and doublons is coupled to spin excitations. Neglect spontaneous doublon production and relaxation. Doublon production due to lattice modulation perturbation Second order perturbation theory. Number of doublons

Propogation of doublons and holes Spectral function: Oscillations reflect shake-off processes of spin waves Comparison of Born approximation and exact diagonalization: Dagotto et al. Hopping creates string of altered spins: bound states