optical lattice emulator strongly correlated systems: from electronic materials to ultracold atoms
TRANSCRIPT
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Optical lattice emulator
Strongly correlated systems: from electronic materials to ultracold atoms
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“Conventional” solid state materialsDescription in terms of non-interacting electrons. Band structure and Landau Fermi liquid theory
First semiconductor transistor
Intel 386DX microprocessor
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“Conventional” solid state materialsElectron-phonon and electron-electron interactions are irrelevant at low temperatures
kx
ky
kF
Landau Fermi liquid theory: when frequency and temperature are smaller than EF electron systems
are equivalent to systems of non-interacting fermions
Ag Ag
Ag
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Non Fermi liquid behavior in novel quantum materials
CeCu2Si2. Steglich et al.,
Z. Phys. B 103:235 (1997)
UCu3.5Pd1.5
Andraka, Stewart, PRB 47:3208 (93)
Violation of the Wiedemann-Franz lawin high Tc superconductorsHill et al., Nature 414:711 (2001)
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Puzzles of high temperature superconductors
Maple, JMMM 177:18 (1998)Unusual “normal” state
Resistivity, opical conductivity,Lack of sharply defined quasiparticles,Signatures of AF, CDW, and SC fluctuations
Mechanism of Superconductivity
High transition temperature,retardation effect, isotope effect,role of elecron-electron and electron-phonon interactions
Competing orders
Role of magnetsim, stripes,possible fractionalization
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Applications of quantum materials:High Tc superconductors
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Picture courtesy of UBC Superconductivity group
High temperature superconductors
Superconducting Tc 93 K
Hubbard model – minimal model for cuprate superconductors
P.W. Anderson, cond-mat/0201429
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Positive U Hubbard model
Possible phase diagram. Scalapino, Phys. Rep. 250:329 (1995)
Antiferromagnetic insulator
D-wave superconductor
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t
U
t
Fermionic atoms in optical lattices
Quantum simulation of the fermionic Hubbard model using ultracold atoms in optical latices
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Fermions in a 3d optical lattice, Kohl et al., PRL 2005
Superfluidity of fermions in an optical lattice, Chin et al., Nature 2006
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Simulation of condensed matter systems: Hubbard Model and high Tc superconductivity
t
U
t
Fermions with repulsive interactions in an optical lattice can be described by the samemicroscopic model as cuprate high temperature superconductorsTheory: Hofstetter et al., PRL 89:220407 (02)
Questions for future work:
• What is the ground state of the Hubbard model away from filling n=1
• Beyond “plain vanilla” Hubbard model a) Boson-Fermion mixtures: Hubbard model + phonons b) Inhomogeneous systems (stripes), role of disorder • Detection of many-body states (spin antiferromagnetisim, d-wave superconductivity , CDW, …)
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How to detect antiferromagnetic order and d-wave pairing in optical lattices?
Quantum noise ?!
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Second order interference from the BCS superfluid
)'()()',( rrrr nnn
n(r)
n(r’)
n(k)
k
0),( BCSn rr
BCS
BEC
kF
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Momentum correlations in paired fermionsGreiner et al., PRL 94:110401 (2005)
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Fermion pairing in an optical lattice
Second Order InterferenceIn the TOF images
Normal State
Superfluid State
measures the Cooper pair wavefunction
One can identify unconventional pairing
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Second order coherence in the insulating state of bosons and fermions
Theory: Altman et al., PRA 70:13603 (2004)
Expt: Folling et al., Nature (2005); Spielman et al., PRL (2007); Rom et al., Nature (2006)
“Bosonic” bunching “Fermionic” antibunching
A powerful tool for detecting antiferromagnetic order
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Boson Fermion mixtures
BEC
Experiments: ENS, Florence, JILA, MIT, ETH, Hamburg, Rice, Duke, Mainz, …
Bosons provide cooling for fermionsand mediate interactions. They createnon-local attraction between fermions
Charge Density Wave Phase
Periodic arrangement of atoms
Non-local Fermion Pairing
P-wave, D-wave, …
Theory: Pu, Illuminati, Efremov, Das, Wang, Matera, Lewenstein, Buchler, …
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Boson Fermion mixtures
“Phonons” :Bogoliubov (phase) mode
Effective fermion-”phonon” interaction
Fermion-”phonon” vertex Similar to electron-phonon systems
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Suppression of superfluidity of bosons by fermions
Similar observation for Bose-Bose mixtures, see Catani et al., arXiv:0706.278
Bose-Fermi mixture in a three dimensional optical lattice Gunter et al, PRL 96:180402 (2006)
See also Ospelkaus et al, PRL 96:180403 (2006)
Issue of heating and density rearrangements need to be sorted out, see e.g. Pollet et al., cond-mat/0609604
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Orthogonality catastrophy due to fermions. Polaronic dressing of bosons.Favors Mott insulating state of bosons
Fermions
Bosons
Fermions
Competing effects of fermions on bosons
Fermions provide screening. Favors superfluid state of bosons
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Quantum regime of bosons
A better starting point:
Mott insulating state of bosons Free Fermi sea
Theoretical approach: generalized Weiss theory
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Weiss theory of the superfluid to Mott transitionof bosons in an optical lattice
Mean-field: a single site in a self-consistent field
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Weiss theory: quantum action
Conjugate variables
Self-consistency condition
Adding fermions
Fermions
Bosons
Screening
Fermions
Orthogonality catastrophy
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SF-Mott transition in the presence of fermionsCompetition of screening and orthogonality catastrophy (G. Refael and ED)
Effect of fast fermions tF/U=5 Effect of slow fermions tF/U=0.7
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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
Same microscopic model