Lecture 2: Ultracold fermions Fermions in optical lattices Fermi Hubbard model Fermions in optical lattices. Fermi Hubbard model. Current state of experiments Lattice modulation experiments Doublon lifetimes Stoner instability Stoner instability
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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
Fermions in optical lattice.Decay of repulsively bound pairsDecay of repulsively bound pairs
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
Fermions in optical lattice.Decay of repulsively bound pairsDecay of repulsively bound 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
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