condensed matter theory at harvard university - lattice...
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![Page 1: Condensed Matter Theory at Harvard University - Lattice ...cmt.harvard.edu/demler/2008_talks/2008_eth.pdfwith fermions in optical lattice. Mott state Related theory work: Kollath et](https://reader036.vdocuments.us/reader036/viewer/2022071507/6128622027d8e4216c014b23/html5/thumbnails/1.jpg)
Lattice modulation experiments
with fermions in optical lattices
and more
Nonequilibrium dynamics of Hubbard model
Ehud Altman Weizmann InstituteDavid Pekker Harvard University
Rajdeep Sensarma Harvard University
Eugene Demler Harvard University
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Antiferromagnetic and superconducting Tc of the order of 100 K
Atoms in optical lattice
Antiferromagnetism and pairing at sub-micro Kelvin temperatures
Fermionic Hubbard modelFrom high temperature superconductors to ultracold atoms
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Fermions in optical lattice
t
U
t
Hubbard model plus parabolic potential
Probing many-body states
Electrons in solids Fermions in optical lattice
• Thermodynamic probesi.e. specific heat
• System size, number of doublons
as a function of entropy, U/t, ω0
• X-Ray and neutron scattering
• Bragg spectroscopy,TOF noise correlations
• ARPES ???
• Optical conductivity• STM
• Lattice modulation experiments
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Outline
• Introduction. Recent experiments with fermions
in optical lattice. Signatures of Mott state
• Lattice modulation experiments in the Mott state.
Linear response theory
• Comparison to experiments
• Lifetime of repulsively bound pairs
• Lattice modulation experiments with d-wave
superfluids
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Mott state of fermions
in optical lattice
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Signatures of incompressible Mott state
Suppression in the number of double occupanciesEsslinger et al. arXiv:0804.4009
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Signatures of incompressible Mott state
Response to external potentialI. Bloch, A. Rosch, et al., arXiv:0809.1464
Radius of the cloud as a functionof the confining potential
Next step: observation of antiferromagnetic order
Comparison with DMFT+LDA models suggests that temperature is above the Neel transition
However superexchange interactions have already been observed
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Radius of the cloud: high temperature expansion
Starting point: zero tunneling.Expand in t/T.
Interaction can be arbitrary
Minimal cloud size for attractive interactionsObserved experimentally
by the Mainz group
Theory: first two terms in t/T expansion
Competition of interaction energy and entropy
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Lattice modulation experiments
with fermions in optical lattice.
Mott state
Related theory work: Kollath et al., PRA 74:416049R) (2006)Huber, Ruegg, arXiv:0808:2350
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Lattice modulation experimentsProbing dynamics of the Hubbard model
Measure number of doubly occupied sites
Main effect of shaking: modulation of tunneling
Modulate lattice potential
Doubly occupied sites created when frequency ω matches Hubbard U
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Lattice modulation experimentsProbing dynamics of the Hubbard model
R. Joerdens et al., arXiv:0804.4009
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Mott state
Regime of strong interactions U>>t.
Mott gap for the charge forms at
Antiferromagnetic ordering at
“High” temperature regime
“Low” temperature regime
All spin configurations are equally likely.Can neglect spin dynamics.
Spins are antiferromagnetically ordered or have strong correlations
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Schwinger bosons and Slave Fermions
Bosons Fermions
Constraint :
Singlet Creation
Boson Hopping
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Schwinger bosons and slave fermions
Fermion hopping
Doublon production due to lattice modulation perturbation
Second order perturbation theory. Number of doublons
Propagation of holes and doublons is coupled to spin excitations.Neglect spontaneous doublon production and relaxation.
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“Low” Temperature
d
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 J, weight by J/t
Incoherent part:dispersion
Propagation of holes and doublons strongly affected by interaction with spin waves
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Propogation of doublons and holes
Spectral function:Oscillations reflect shake-off processes of spin waves
Hopping creates string of altered spins: bound states
Comparison of Born approximation and exact diagonalization: Dagotto et al.
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“Low” Temperature
Rate of doublon production
• Low energy peak due to sharp quasiparticles
• Broad continuum due to incoherent part
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“High” Temperature
Atomic limit. Neglect spin dynamics.All spin configurations are equally likely.
Aij (t’) replaced by probability of having a singlet
Assume independent propagation of doublons and holes.Rate of doublon production
Ad(h) is the spectral function of a single doublon (holon)
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Propogation of doublons and holesHopping creates string of altered spins
Retraceable Path Approximation Brinkmann & Rice, 1970
Consider the paths with no closed loops
Spectral Fn. of single hole Doublon Production Rate Experiments
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Ad(h) is the spectral function of a single doublon (holon)
Sum Rule :
Experiments:Possible origin of sum rule violation
The total weight does not scale
quadratically with t
• Nonlinearity
• Doublon decay
Lattice modulation experiments. Sum rule
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Lattice modulation experimentsProbing dynamics of the Hubbard model
R. Joerdens et al., arXiv:0804.4009
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Doublon decay rateinspired by experiments in ETH
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Energy Released ~ U
�Energy carried by
spin excitations
~ J =4t2/U
� Relaxation requires
creation of ~U2/t2
spin excitations
Relaxation of doublon hole pairs in the Mott state
Relaxation rate
Large U/t : Very slow Relaxation
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Alternative mechanism of relaxation
LHB
UHB
µ
• Thermal escape to edges
• Relaxation in compressible edges
Thermal escape time
Relaxation in compressible edges
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Doublon decay in a compressible state
How to get rid of the excess energy U?
Compressible state: Fermi liquid description
Doublon can decay into apair of quasiparticles with
many particle-hole pairs
U
p-p
p-h
p-h
p-h
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Doublon decay in a compressible state
Decay amplitude
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Doublon decay in a compressible state
Fermi liquid description
Single particle states
Doublons
Interaction
Decay
Scattering
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Doublon decay in a compressible state
Decay rate contained
in self-energy
Self-consistent equations for doublon
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Doublon decay in a compressible state
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Lattice modulation experiments
with fermions in optical lattice.
Detecting d-wave superfluid state
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• consider a mean-field description of the superfluid
• s-wave:
• d-wave:
• anisotropic s-wave:
Setting: BCS superfluid
Can we learn about paired states from lattice modulation
experiments? Can we distinguish pairing symmetries?
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Modulating hopping via modulation
of the optical lattice intensity
Lattice modulation experiments
where
-3 -2 -1 0 1 2 3
-3
-2
-1
0
1
2
3
• Equal energy
contours
Resonantly exciting quasiparticles with
Enhancement close to the bananatips due to coherence factors
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Distribution of quasi-particles
after lattice modulation
experiments (1/4 of zone)
Momentum distribution of
fermions after lattice modulation
(1/4 of zone)
Can be observed in TOF experiments
Lattice modulation as a probe of d-wave superfluids
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number of quasi-particles density-density correlations
• Peaks at wave-vectors connecting tips of bananas
• Similar to point contact spectroscopy
• Sign of peak and order-parameter (red=up, blue=down)
Lattice modulation as a probe of d-wave superfluids
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Scanning tunneling spectroscopy of high Tc cuprates
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Conclusions
Experiments with fermions in optical lattice open
many interesting questions about dynamics of the
Hubbard model
Thanks to:
Harvard-MIT