Transcript Ultracold fermions in optical lattices

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
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
SU(N) Magnetism with
Ultracold Alkaline-Earth Atoms
A. Gorshkov, et al., Nature Physics 2010
Ex: 87Sr (I = 9/2)
Alkaline-Earth atoms in optical lattice:
|e> = 3P0
698 nm
150 s ~ 1 mHz
|g> = 1S0
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
Non-local
pairing
of holes
Leading istability:
d-wave
Scalapino et al, PRB (1986)
-k’
k’
k
spin
fluctuation
-k
Pairing of holes
in the Hubbard model
BCS equation for pairing amplitude
Q
-k’
k’
-
+
+
-
dx2-y2
k
spin
fluctuation
-k
Systems close to AF instability:
c(Q) is large and positive
Dk 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
n=1
D-SC
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’)
kF
k
n(r)
BCS
BEC
Dn(r, r' )  n(r)  n(r' )
Dn(r,r) BCS  0
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
current
experiments
Mott
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
TN
current
experiments
Mott
U
Negative U Hubbard model
|U|
t
t
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
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Anomalous expansion:
Competition of energy and entropy
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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
“Low” Temperature
Schwinger bosons Bose condensed
d
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 t2/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
“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)
Spectral Fn. of single hole
Theory:
Sensarma et al.,
PRL 103, 035303 (2009)
Temperature dependence
Psinglet
Density
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
=4t2/U
 Relaxation requires
creation of ~U2/t2
spin excitations
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
p -h
p -h
Doublon can decay into a
pair of quasiparticles with
many particle-hole pairs
U
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)
V0 tunable by the magnetic field
Can tune through bound state
scattering length
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 (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
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)
Earlier work by C. Salomon et al., 2003
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)
V0 tunable by the magnetic field
Can tune through bound state
scattering length
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
p
k
-k
-p
Microscopic Hamiltonian
Schrödinger equation
T-matrix
Lippman-Schwinger equation
k
k
-k
p
-p
k
p
-k
-p
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
Need to make sure that we do not
include interaction effects on the Fermi
liquid state in scattered state energy
p
k
-k
-p
Cooperon
Grand canonical ensemble
Define
Cooperon equation
Cooperon vs T-matrix
k
k
-k
p
-p
k
p
-k
-p
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
Effects of finite
temperature
Three body recombination
as in Shlyapnikov et al., 1996;
Petrov, 2003; Esry 2005
From wide to
narrow resonances
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
Bosons
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