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Quantum%20Monte%20Carlo%20methods

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Title: Quantum%20Monte%20Carlo%20methods


1
  • Quantum Monte Carlo methods
  • applied to ultracold gases
  • Stefano Giorgini
  • Istituto Nazionale per la Fisica della Materia
    Research and Development Center
    on Bose-Einstein Condensation
  • Dipartimento di Fisica UniversitĂ  di Trento

BEC CNR-INFM meeting 2-3 May 2006
2
  • QMC simulations have become an important tool in
    the study of dilute ultracold gases
  • Critical phenomena
  • Shift of Tc in 3D GrĂĽter et al. (97), Holzmann
    and Krauth (99), Kashurnikov et al. (01)
  • Kosterlitz-Thouless Tc in 2D Prokofev et al.
    (01)
  • Low dimensions
  • Large scattering length in 1D and 2D Trento (04
    - 05)
  • Quantum phase transitions in optical lattices
  • Bose-Hubbard model in harmonic traps Batrouni et
    al. (02)
  • Strongly correlated fermions
  • BCS-BEC crossover Carlson et al. (03), Trento
    (04 - 05)
  • Thermodynamics and Tc at unitarity Bulgac et
    al. (06), Burovski et al. (06)

3
  • Continuous-space QMC methods
  • Zero temperature
  • Solution of the many-body Schrödinger equation
  • Variational Monte Carlo
  • Based on variational principle
  • energy upper bound
  • Diffusion Monte Carlo
  • exact method for the ground state of Bose
    systems
  • Fixed-node Diffusion Monte Carlo (fermions and
    excited states)
  • exact for a given nodal surface ? energy
    upper bound
  • Finite temperature
  • Partition function of quantum many-body system
  • Path Integral Monte Carlo
  • exact method for Bose systems

4
  • Low dimensions large scattering length

5
  • 1D Hamiltonian
  • if g1D large and negative (na1Dltlt1) metastable
    gas-like state of hard-rods of size a1D

g1Dgt0 Lieb-Liniger Hamiltonian (1963) g1Dlt0
ground-state is a cluster state
(McGuire 1964)
Olshanii (1998)
at na1D ? 0.35 the inverse compressibility
vanishes
gas-like state rapidly disappears forming clusters
6
  • Correlations are stronger than in the
    Tonks-Girardeau gas
  • (Super-Tonks regime)

Power-law decay in OBDM
Peak in static structure factor
Breathing mode in harmonic traps
TG
mean field
7
  • Equation of state of a 2D Bose gas

Universality and beyond mean-field effects
  • hard disk
  • soft disk
  • zero-range

for zero-range potential mc20 at
na2D2?0.04 onset of instability for cluster
formation
8
  • BCS-BEC crossover in a Fermi gas at T0

-1/kFa
BCS
BEC
9
Equation of state
beyond mean-field effects confirmed by study of
collective modes (Grimm)
BEC regime gas of molecules mass 2m - density
n/2 scattering length am
am0.6 a (four-body calculation of Petrov et
al.) am0.62(1) a (best fit to FN-DMC)
10
  • Frequency of radial mode (Innsbruck)

QMC equation of state
Mean-field equation of state
11
  • Momentum distribution

JILA in traps
Condensate fraction
12
  • Static structure factor (Trento Paris ENS
    collaboration)
  • ( can be measured in Bragg scattering experiments)

at large momentum transfer kF ?? k ??
1/a crossover from S(k)2 free
molecules to S(k)1 free atoms
13
  • New projects
  • Unitary Fermi gas in an optical lattice (G.
    Astrakharchik Barcelona)
  • d1/q?/2? lattice spacing
  • Filling 1 one fermion of each spin component
    per site (ZĂĽrich)
  • Superfluid-insulator transition
  • single-band Hubbard Hamiltonian is inadequate

14
(No Transcript)
15
  • S1

S20
16
  • Bose gas at finite temperature (S. Pilati
    Barcelona)
  • Equation of state and universality

T ? Tc
T ? Tc
17
  • Pair-correlation function and bunching effect
  • Temperature dependence of condensate fraction
    and superfluid density
  • ( N. Prokofevs help on implemention of
    worm-algorithm)

T 0.5 Tc
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