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Quantum Monte Carlo methods
applied to ultracold gases
Stefano Giorgini
BEC CNR-INFM meeting 2-3 May 2006
Istituto Nazionale per la Fisica della Materia
Research and Development Center on
Bose-Einstein Condensation
Dipartimento di Fisica – Università di Trento
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 Prokof’ev 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)
Continuous-space QMC methods
Zero temperature
• Solution of the many-body Schrödinger equation
Variational Monte Carlo
Based on variational principle
energy upper bound
E
T H T
T T
where  T trial function
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
Low dimensions + large scattering length
1D Hamiltonian
H 1D
2 N 2

 g1D   ( zij )

2
2m i 1 zi
i j
g1D  
2 a3 D 
a3 D 
2

1

1
.
03


2
ma1D
ma 
a 
2
2
g1D>0 Lieb-Liniger Hamiltonian (1963)
g1D<0 ground-state is a cluster state
(McGuire 1964)
1
Olshanii (1998)
if g1D large and negative (na1D<<1) metastable gas-like state of hard-rods of size a1D
EHR  2  2 n 2
1

N
6m (1  na1D ) 2
at na1D  0.35 the inverse
compressibility vanishes
gas-like state rapidly disappears
forming clusters
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
Equation of state of a 2D Bose gas
Universality and beyond mean-field effects
EMF 2 2
n

N
m ln(1 / na22D )
• hard disk
• soft disk
• zero-range
for zero-range potential
mc2=0 at na2D20.04
onset of instability for cluster formation
BCS-BEC crossover in a Fermi gas at T=0
-1/kFa
BEC
BCS
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]
E / N  b / 2
F

5
128


3/ 2
k F am 1 
(
k
a
)

...
F m

3
18
 15 6

am=0.6 a (four-body calculation of Petrov et al.)
am=0.62(1) a
(best fit to FN-DMC)
Frequency of radial mode (Innsbruck)
QMC
equation of state
Mean-field
equation of state
Momentum distribution
JILA in traps
Condensate fraction
n0  1 
8
3 
(nm am3 )1/ 2
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
New projects:
• Unitary Fermi gas in an optical lattice (G. Astrakharchik + Barcelona)
 2q 2
sin 2 (qx)  sin 2 (qy)  sin 2 (qz)
Vext (r )  s
2m
d=1/q=/2 lattice spacing
Filling 1: one fermion of each spin component per site (Zürich)
Superfluid-insulator transition
single-band Hubbard Hamiltonian is inadequate
1.0
superfluid fraction
condensate fraction
0.8
0.6
0.4
0.2
0.0
0
1
2
s
3
S=1
S=20
• Bose gas at finite temperature (S. Pilati + Barcelona)
Equation of state and universality
T  Tc
T  Tc
Pair-correlation function and bunching effect
Temperature dependence of condensate fraction and superfluid density
(+ N. Prokof’ev’s help on implemention of worm-algorithm)
T = 0.5 Tc