Quantum Monte Carlo methods

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• 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 Università di Trento

BEC CNR-INFM meeting 2-3 May 2006
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• QMC simulations have become an important tool in
  the study of dilute ultracold gases
• Critical phenomena
• Shift of Tc in 3D Grüter 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)

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• 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

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• Low dimensions large scattering length

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• 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
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• 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
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• 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
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• BCS-BEC crossover in a Fermi gas at T0

-1/kFa
BCS
BEC
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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)
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• Frequency of radial mode (Innsbruck)

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

JILA in traps
Condensate fraction
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• 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
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• 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 (Zürich)
• Superfluid-insulator transition
• single-band Hubbard Hamiltonian is inadequate

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• S1

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

T ? Tc
T ? Tc
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• 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