Download r - Barcelona Quantum Monte Carlo group

Many-body physics with
atomic gases in optical lattices
Sebastiano Pilati
“Abdus Salam” International Centre for Theoretical Physics
Trieste- Italy
Theory:
Ping Nang Ma
ETH – Zurich
Prof. Matthias Troyer
ETH – Zurich
Prof. Stefano Giorgini
INO-CNR “BEC” Center – University of Trento
Prof. Xi Dai
Chinese Academy of Science – Beijing
Barcelona – 19 January 2012
Experiments:
Prof. Salomon group
ENS - Paris
Prof. Nägerl group
University of Innsbruck
Main idea:
use ultra-cold gases to test/further develop
computational techniques for many-fermion systems
Big challenge: strongly correlated electronic systems
quantum magnetism (ferromagnetism, antif., spin liquids)
giant magnetoresistance
High-TC superconductors
Motivation:
ultracold gases are accessible and tunable
Measuring the Equation of State:
-) Exp:
Nascimbene, Navon, Jiang, Chevy, Salomon, Nature 2010 (ENS - Paris)
-) Theory: Ho, Zhou, Nature Phys (2009)
Tuning the interaction strength
SPIN UP:
SPIN DOWN:
s-wave scattering length
Feshbach resonance
“metastable”
repulsive branch
magnetic field
attractive branch
BEC-BCS crossov.
BEC
unitarity
BCS
Theory:
Legget, Nozier-Schmitt-Rink
Exp: ENS(Paris), MIT etc.
EOS of attractive Fermi gas:
BEC-BCS crossover
 Quantum MC
—— experiment
weakly attractive
Fermi gas
weakly repulsive
Bose gas of molecules
interaction parameter
BEC
unitarity
BCS
QMC: Astrakharchik at al. PRL (2004)
SP, Giorgini PRL (2008)
Exp: Nascimbene at al. Nature(2010) (ENS-Paris)
Spin-imbalanced Fermi gases:
N↑ > N↓
EF↑
EF↓
?
Polarized superfluid
Phase-separated state
magnetic
susceptibility
normalized pressure
▬▬ superfluid
▬▬ normal
 exp.
▬▬ QMC (normal)
▬▬ QMC (SF)
interaction parameter 
b = (μ↑ - μ ↓) / μ
BCS
Energy difference
EN-ES [3/5NEF]
normalized pressure
Equation of state: normal phase
  / 2m a
BEC
▬▬ QMC
3 2
EC  E N  ES 
N
8 EF
Nascimbène, Navon, Pilati, Chevy, Giorgini, Georges, Salomon, PRL 106 215303 (2011)
MIT 2008
Tuning the interaction strength
SPIN UP:
SPIN DOWN:
s-wave scattering length
Feshbach resonance
“metastable”
repulsive branch
magnetic field
attractive branch
BEC-BCS crossov.
BEC
unitarity
BCS
Theory:
Legget, Nozier-Schmitt-Rink
Exp: ENS(Paris), MIT etc.
Two component repulsive Fermi gases
STONER INSTABILITY → paradigm for itinerant ferromagnetism
SPIN UP:
SPIN DOWN:
 N↑ and N↓ fixed
interaction strength
 PERTURBATIVE THEORIES:
Mean-field theory (Stoner model):
2nd order phase t. at kFa = 1.57
2nd order perturbation theory (Duine and MacDonald,PRL2009):
1st order phase t. at kFa = 1.054
 EXPERIMENT@MIT:
Repulsive Fermi gases: Equation of State with Fixed-Node DMC
“Fully polarized” ferromagnet

Hard-Sphere
▲
Soft-Sphere

Square-Well
——
2nd
order p. th.
Eint = 0
Ekinetic = 22/3 EIFG
 Small non-universal effects at kFa > 0.6
 Homogeneous phase unstable at kFa ≈ 0.9
→ but repulsive gas is unstable!
(see exp. @ MIT : Sanner et al. arXiv:1108.2017)
Inverse magnetic susceptibility
SW
HS
Ferromagnetic transition
kFa = 0.82(1) (HS) kFa = 0.86(1) (SW)
Periodic potentials:
optical lattices
V0
d /2
V ( x)  V0 sin 2 (x / d )
 harmonic terms
V0 : laser intensity
λ : laser wavelength
Bosons: single-band Hubbard model
Wannier w0
phase diagram
Fisher at al. (1989)
Single-atom (and single-site) microscope
I. Bloch’s group –Munich (2011)
How to describe inter-atomic interactions:
dilute systems →
only s-wave scattering length matters
V(r)
V(r)
r
r
1. “BEST” approx. : Fermi-Huang Contact pseudopotential
4 2
V r  
a r  r r
m
2.
(regularized δ-function interaction)
 gas-like state is out of equilibrium
r0 << a
 needs huge number of Bloch bands
 effective (low-energy) Hubbard parameter U H.P. Büchler PRL (2010)
δ-function interaction → 1st Born approximation
4 2
V r  
a r 
m
3. Hard-core potential (purely repulsive)
V r    if r  a, 0 otherwise
 used to determine Hubbard parameter U
Jaksch et al. PRL (1998)
 gas-like state is at equilibrium
r0 >> a
r0 = a
 possible non-universal effects at very large a
Hard Spheres in OL (cont. space)
vs. single-band Bose-Hubbard
S.P., Troyer, arXiv: 1108.1408
optical lattice intensity
n=1
Insulator

——
♦
——
hard-spheres in OL (cont. space)
Single-band Bose-Hubbard
Exp. (Innsbruck)
Multi-orbital BH (Luehmann et al.)
Superfluid
freezing density
Interaction strength
• single-band Bose-Hubbard model: Capogrosso-Sansone et al. PRB 75 134302 (2007)
• Hard-sphere freezing: Kalos et al. PRA 9 2178 (1974)
Fermi gases in OL: \quantum magnetism
ferromagnetic phase
anti-ferromagnetic phase
→ (strong) repulsive interactions + periodic potential
Feshbach resonances
Optical lattices
Tuning the optical lattice intensity
Unexplored regime:
Kohn-Sham density functional
theory
Homogeneous system
QMC predicts itinerant ferromagnetism at
very large interaction strength (kFa ≈0.9)
Chang et al. PNAS (2011), SP,Bertaina et al. PRL(2010),
Conduit et al. PRL(2009)
Molecule formation is too fast!
Pekker et al. PRL(2011)
–
see D. Pekker’s talk
Sanner, arXiv:1108.2017 (MIT)
Single-band lattice models
• Metal - Mott insulator transition observed
Esslinger’s group (ETH) : Jördens et al. Nature (2008)
Bloch’s group: Schneider et al. Science (2008)
• anti-ferromagnetic order not observed yet!
Kohn-Sham Density Functional Theory
the standard computational method in material science
1998 Nobel prize in chemistry
Walter Kohn
Es.: band structure of silicon
John A. Pople
successful in calculating properties of
many metals, insulators, semiconductors
wrong gap in transition metal compounds
→ simplified models (Hubbard)
Chelikowsky and Cohen, PRB (1974)
Use of DFT in quantum chemistry and materials science
DFT: reduces the N-body problem to a 1-body problem
Energy-density functional:
E   EKIN     drV r  r    drdrVINT  r  r  r  r  EXC  
kinetic energy
external potential
mean-field interaction
Basic approx. → Local Density Approximation:
note: not the AMO LDA!!
eXchange and Correlation
(unknown!)
homo
 (r) 
EXC     dr XC
• 3D electron gas : Ceperley, Alder 1980
• 3D repulsive Fermi gas: this work
From Fixed-Node Quantum Monte Carlo
Beyond LDA: GGA, meta GGA, hybrid functionals, LDA + U, GW, LDA + DMFT, LDA +QMC...
IDEA:
→ use KS-DFT in shallow optical lattices
→ test new energy-density functionals using
ultracold gases in deep OL: clean and tunable
KS-DFT with LDA of a repulsive Fermi gas in OL
 Fixed-Node Diffusion MC simulation of the homogeneous Fermi gas with short-range interaction
 Accurate parametrization of the EOS:
Energy vs. P
E = E(kFa, P)
Energy vs. kFa and magnetic susc.
Polaron energy and eff. mass
• Non universal corrections: small for kFa < 1
• Hartree term is local: non self-interaction error
• No core electrons – no pseudopotentials
KS-DFT with LDA for a repulsive Fermi gas in OL
GROUND-STATE PHASE DIAGRAM
part. pol. ferrom.
fully pol.ferrom.
param.
param.
increasing OL intensity
TECHNICAL DETAILS: exchange-interaction functional from QMC
solve self-consistent Kohn-Sham eq.
Bloch Ansatz (203 k-points)
expanded in plane wave basis (113 states)
n=1
ferrom.
anti-ferrom.
staggered polarization – uniform polarization
Phase diagram at half-filling
Band-structure
spin-density wave gap
Fermi gases in OL:
QMC vs. KF-DFT (preliminary)
n = 0.5
a/d = 0.04
SUMMARY:
KS – DFT
→ extensive toolbox to simulate Fermi gases in OL
experiments with OL → test bed to develop new energy density functionals
(maybe more important that solving Hubbard model!)
KS-DFT for repulsive Fermi gases in OL:
• adding an OL favours ferromagnetism
• KS-LDA predicts anti-ferromagnetism at n = 1 and T = 0
OUTLOOK:
• finite temperature DFT
• attractive Fermi gases in OL (BEC-BCS crossover)
→ superconducting DFT
(Gross et al. PRL 1988, A. Bulgac PRA 2007)
• time-dependent DFT (see TD-DFT for unitary Fermi gas Bulgac et al. 2011)
• beyond LDA: GGA, hybrid functionals, LDA +U, LDA+GW, LDA + DMFT, LDA + QMC...
ADDITIONAL MATERIAL
→
Two-body wave functions
Kohn-Sham Density Functional Theory (with LDA and beyond):
the standard computational method in material science
 very accurate for weakly/moderately correlated materials
 wrong gap in transition metal compounds → simplified models (Hubbard)
IDEA:
→ use KS-DFT in shallow optical lattices
→ develop energy-density functionals
for the strongly correlated regime
NOTE: we can tune independently interaction strength and external potential
Band Structure
a = 0.08d
E [ER]
E [ER]
a = 0.16d
E [ER]
E [ER]
V0 = 4ER
E [ER]
a = 0.04d
E [ER]
V0 = 2ER