fermions in optical lattices fermi hubbard...

Lecture 2: Ultracold fermions

Fermions in optical lattices Fermi Hubbard modelFermions in optical lattices. Fermi Hubbard model.Current state of experiments

Lattice modulation experiments

Doublon lifetimes

Stoner instabilityStoner instability

Ultracold fermions in optical latticesUltracold fermions in optical lattices

Fermionic atoms in optical latticesp

U

t

U

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

Quantum simulations with ultracold atoms

Atoms in optical lattice

Antiferromagnetic and superconducting Tc

Atoms in optical lattice

Antiferromagnetism and pairing at sub-micro Kelvin p g

of the order of 100 K temperatures

Same microscopic model

Positive U Hubbard modelPositive U Hubbard modelPossible phase diagram. Scalapino, Phys. Rep. 250:329 (1995)

Antiferromagnetic insulator

D-wave superconductor? p

-?

-

Repulsive Hubbard model at half-filling

currentexperimentsTN experimentsN

Mott statewithout AF order

U

Signatures of incompressible Mott state of fermions in optical latticestate of fermions in optical lattice

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

Compressibility measurementsU. Schneider et al., Science (2008)R. Joerdens et al., Nature (2008) U. Schneider et al., Science (2008)

Lattice modulation experiments with fermions in optical latticewith fermions in optical lattice.

Probing the Mott state of fermions

Related theory work: Kollath et al., PRL (2006)

Sensarma, Pekker, Lukin, Demler, PRL (2009)

y ( )Huber, Ruegg, PRB (2009)Orso, Iucci, et al., PRA (2009)

Lattice modulation experimentsP bi d i f th H bb d d lProbing dynamics of the Hubbard model

Modulate lattice potentialModulate lattice potential

Measure number of doubly occupied sites

M i ff t f h ki d l ti f t liMain effect of shaking: modulation of tunneling

Doubly occupied sites created when frequency matches Hubbard U

Lattice modulation experimentsP bi d i f th H bb d d lProbing 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 atMott gap for the charge forms at

Antiferromagnetic ordering at

“High” temperature regimeAll spin configurations are equally likely

“L ” t t i

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

“Low” temperature regimeSpins are antiferromagnetically ordered or have strong correlationsg

Schwinger bosons and Slave Fermions

Bosons Fermions

Constraint :

Singlet Creation

B H iBoson Hopping

Schwinger bosons and slave fermionsFermion hoppingFermion 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 doublonsSecond order perturbation theory. Number of doublons

S h i b B d d

“Low” Temperature

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

Schwinger bosons Bose condensed

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:Sharp coherent part:dispersion set by t2/U, weight by t/U

Incoherent part:dispersiondispersion

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 partp

• Spin wave shake-off peaks

“High” TemperatureCalculate self-energy of doublons and holes interacting with incoherentspin excitations (Schwinger bosons) in the non-crossing approximation

Sensarma et al., PRL (2009)Tokuno et al. arXIv:1106.1333

Equivalent to Retraceable Path Approximation Brinkmann & Rice, 1970

In calculating spectral function consider paths with no closed loops

“High” TemperatureSpectral Fn. of single hole

Doublon production rateExperiment: R. Joerdens et al.,

Doublon production rate

,Nature 455:204 (2008)

Theory: Sensarma et al PRL (2009)

d

h Sensarma et al., PRL (2009)Tokuno et al. arXIv:1106.1333

h

Temperature dependenceTemperature dependence

Reduced probability to find a singlet on neighboring sites

nsit

y

ngle

tRadius Radius

Den P s

inRadius Radius

D. Pekker et al., upublished

Fermions in optical lattice.Decay of repulsively bound pairsDecay of repulsively bound pairs

Experiments: ETH groupExperiments: ETH groupTheory: Sensarma, Pekker, et. al.

Ref: N. Strohmaier et al., PRL 2010


Doublons – repulsively bound pairsp y pWhat is their lifetime?

Direct decay isnot allowed byenergy conservation

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

Decay of doublon into a pair of quasiparticles req ires creation of man particle hole pairsrequires creation of many particle-hole pairs


Experiments: N. Strohmaier et. al.

Relaxation of doublon- hole pairs in the Mott state

Energy U needs to be b b d babsorbed by

spin excitations

Energy carried by Relaxation requires

spin excitations

~ J =4t2/U

creation of ~U2/t2

spin excitations~ J =4t2/U

Relaxation rate Sensarma et al., PRL 2011

Very slow, not relevant for ETH experiments

Doublon decay in a compressible stateExcess energy U isconverted to kineticenergy of single atoms

Compressible state: Fermi liquid description

p-h

Doublon can decay into aU p-h

p-h

pair of quasiparticles with many particle-hole pairs

Up-p

p h

Doublon decay in a compressible statePerturbation theory to order n=U/6tDecay probability

Doublon Propagator Interacting “Single” Particles

Doublon decay in a compressible stateN. Strohmaier et al., PRL 2010

Expt: ETHZTheory: Harvard

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

Why understanding doublondecay rate is interesting

Important for adiabatic preparation of strongly correlated systems in optical lattices

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

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 matterAnalogy to pump and probe experiments in condensed matter systems

Doublon relaxation in organic gMott insulators ET-F2TCNQ

One dimensional Mott insulator ET-F2TCNQ

t=0.1 eV U=0.7 eV

Photoinduced metallic stateH. Okamoto et al., PRL 98:37401 (2007)S. Wall et al. Nature Physics 7:114 (2011)

Surprisingly long relaxation time 840 fsg y g

h/t = 40 fs

Photoinduced metallic stateH. Okamoto et al., PRL 98:37401 (2007)S. Wall et al. Nature Physics 7:114 (2011)

1400 ft=0.1 eVw=4t=0.4eV

1400 fscomparable to experimentally

d 840U=0.7 eV measured 840 ms

Exploring beyond simple Hubbard modelsimple Hubbard modelwith ultracold fermions

SU(N) Hubbard model withUltracold Alkaline-Earth AtomsUltracold Alkaline Earth Atoms

Ex: 87Sr (I = 9/2)

Theory: A. Gorshkov, et al., Nature Physics 2010

( )

|e = 3P0698 nm

150 s ~ 1 mHz

Experiments: realization of SU(6) fermionsTakahashi et al. PRL (2010)

|g = 1S0

Nuclear spin decoupled from electrons SU(N=2I+1) symmetry

Also J. Ye et al., Science (2011)

bit l d f f d i bit l h i

Nuclear spin decoupled from electrons SU(N 2I+1) symmetry→ SU(N) Hubbard 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) K d l i d l [f N 2 l l i i→ SU(N) Kondo lattice model [for N=2, colossal magnetoresistance in manganese oxides and heavy fermion materials]

Nonequilibrium dynamics: expansion of interacting f i i ti l l ttifermions in optical lattice

U. Schneider et al., Nature Physics 2012

New dynamical symmetry:id ti l l d f iidentical slowdown of expansionfor attractive and repulsiveinteractions

Competition between pairing and f i i bili i iferromagnetic instabilities in ultracold Fermi gases near Feshbach resonances

Phys. Rev. Lett. 2010

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

Motivated by experiments of G.‐B. Jo et al., Science (2009)

Stoner model of ferromagnetismSpontaneous spin polarizationSpontaneous spin polarizationdecreases interaction energybut increases kinetic energy ofelectronselectrons

Mean‐field criterion

U N(0) = 1( )

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

then

Kanamori’s counter‐argument: renormalization of U.

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 wereExperiments weredone dynamically.What are implicationsof dynamics?yWhy spin domains could not be observed?Earlier work by C. Salomon et al., 2003

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

F hb h h i b d ff i l iFeshbach physics beyond effective repulsive interaction

Feshbach resonance

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

Example:

scattering lengthV(x)

V t bl b th ti fi ldV0 tunable by the magnetic fieldCan tune through bound state

Feshbach resonancel b dTwo particle bound state

formed in vacuum

BCS instabilityStoner instability

Molecule formationand condensation

This talk: Prepare Fermi state of weakly interacting atoms.p y gQuench to the BEC side of Feshbach resonance.System unstable to both molecule formationand Stoner ferromagnetism. Which instability dominates ?

Pair formation

Two‐particle scattering in vacuum

k ‐kp

k ‐k

Microscopic Hamiltonian‐p

p

Schrödinger equationSchrödinger equation

T-matrix

Lippman-Schwinger equation

kk k pp k pp’

-k -k -p’-p -p -p

On-shell T-matrix. Universal low energy expression

For positive scattering length bound state atappears as a pole in the T-matrix

Cooperonl h fTwo particle scattering in the presence of a Fermi sea

Need to make sure that we do not

k

p include interaction effects on the Fermi liquid state in scattered state energy

‐k

‐p

Cooperon vs T-matrix

kk k pp k pp’

-k -k -p’-p -p

p

-p

Cooper channel response functionLinear response theory

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

Thi t li i F i tThis argument applies in vacuum. Fermi sea preventsformation of real Feshbach molecules by Pauli blocking.

Molecule Fermi sea

Pairing instabilityTime dependent variational wavefunction

Time dependence of uk(t) and vk(t) due to BCS(t)

For small BCS(t):

Pairing instability

From wide to narrow resonances

Effects of finite temperaturetemperature

Three bod recombinationThree body recombination as in Shlyapnikov et al., 1996;Petrov, 2003; Esry 2005

Observed in recent experimentsby Grimm’s group, arXiv:1112.0020

Magnetic instability

Stoner instability. Naïve theory

Linear response theory

Spin response function

S i ll ti d i b th l f f tiSpin 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 functionSelf-consistent equation on response function

Spin susceptibility for non-interacting gas

RPA expression for the spin response functionthe spin response function

Quench dynamics across Stoner instability

Stoner criterion

For U>Uc unstable collective modes

Unstable modes determine characteristic lengthscalegof magnetic domains

Stoner quench dynamics in D=3

Scaling near transition

Growth rate of magnetic domainsgUnphysical divergenceof the instability rate at unitarity

Domain size

Stoner instability

= + + + …= + + + …

Stoner instability is determined by two particlescattering amplit de

Divergence in the scattering amplitude arises f b d t t f ti B d t t i

scattering amplitude

from bound state formation. Bound state is strongly affected by the Fermi sea.

Stoner instabilityRPA spin susceptibility

Interaction = Cooperon

Stoner instability

Pairing instability always dominates over pairing

If ferromagnetic domains form they form at large qIf ferromagnetic domains form, they form at large q

Tests of the Stoner magnetism

Spin fluctuations relative to Extremely fast molecule formation

C. Sanner et al., arXiv:1108.2017

Spin fluctuations relative tononinteracting fermions

Extremely fast molecule formation rate at short times

Only short range correlationsand no domain formation

Atoms loss rate is 13% of EFon resonance (averaged over trap)

Lecture 2: Ultracold fermions

Fermions in optical lattices Fermi Hubbard modelFermions in optical lattices. Fermi Hubbard model.Current state of experiments

Lattice modulation experiments

Doublon lifetimes

Stoner instabilityStoner instability

Future directions in ultracold atomsFuture directions in ultracold atomsNonequilibrium quantum many-body dynamics

Long intrinsic time scales- Interaction energy and bandwidth ~ 1kHzInteraction energy and bandwidth 1kHz- System parameters can be changed over this time scale

Decoupling from external environment- Long coherence times

Can achieve highly non equilibrium quantum many-body states

Emergent phenomena in dynamics of classical systems

Universality in quantum many-body systems in equilibrium

Solitons in nonlinear wave propagation Broken symmetries

B d ll i th f T di t F i li id t tBernard cells in the presence of T gradient Fermi liquid state

Do we have emergent universal phenomena in nonequilibrium dynamics of many-body

quantum systems?

Hubbard model at half filling

TN

Paramagnetic Mott phase:

paramagnetic

one fermion per sitecharge fluctuations suppressedno spin order

Mott phasep

UBCS-typetheory applies

Heisenbergmodel applies

Hubbard model at half filling

TN

currentexperimentsexperiments

UBCS-typetheory applies

Heisenbergmodel applies

fermions in optical lattices fermi hubbard...

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