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Properties of strongly correlated Fermi gas
from Quantum Monte Carlo – constraints on
Energy Density Functional

Piotr Magierski
Warsaw University of Technology & Univ. of Washington

Collaborators:
A. Bulgac
J.E. Drut
M.M. Forbes
K.J. Roche
G. Wlazłowski

- University of Washington
- University of North Carolina
- Washington State Univ.
– Pacific Northwest National Lab.
- Warsaw Univ. of Techn.

What is a unitary gas?

A gas of interacting fermions is in the unitary regime if the average
separation between particles is large compared to their size (range of
interaction), but small compared to their scattering length.

n r03 << 1 n |a|3 >> 1

n - particle density
a - scattering length
r0 - effective range
NONPERTURBATIVE
REGIME

System is dilute but
strongly interacting!
Universality:

E  x     x  EFG

 (0)  0.37(1) -

EFG

; x= T

F

Exp. estimate

- Energy of noninteracting Fermi gas

BCS – BEC crossover
Eagles (1969), Leggett (1980): Variational approach

gs    u k +v k aˆk† aˆ† k   vacuum

BCS wave function

k

1

BCS limit: k F as

 

BEC limit:

1

kF a s

 

as - scattering length

  F


  
8

exp


F
e2
2
k
a
 F s

chemical potential

 

pairing gap



Usual BCS solution for small and
negative scattering lengths, with
exponentially small pairing gap
describing the system of spatially
overlapping Cooper pairs.

2

2mas 2

= 

Eb
2

4 F
3 k F as

Gas of weakly repelling molecules
with binding energy Eb, essentially
all at rest (almost pure BEC state)

No singularity within the whole range of scattering length!
Smooth crossover from spatially overlapping Cooper pairs to
tightly bound difermionic molecules
Beyond mean field: Nozieres, Schmitt-Rink (1985), Randeria et al.(1993)

Thermodynamics of the unitary Fermi gas
ENERGY:

3
T
E ( x)   ( x) F N ; x 
5
F

S ( x) 3  '( y )
ENTROPY/PARTICLE:  ( x) 
 
dy
N
50 y
x

3
FREE ENERGY: F  E  TS   ( x) F N
5
 ( x)   ( x)  x ( x)
E 2
N
PRESSURE: P  
  ( x) F
V 5
V
2
PV  E
3

Note the similarity to
the ideal Fermi gas

Universal Tan relations

C
lim n(k )  4 ,
k 
k

1  k  1 , C - contact
a
r0

- Contact measures the probability that two fermions of opposite
spins are close together.

E

d 3k




 2 
1

2
k2 
C
Total energy
n
(
k
)


C
 

of the system
2m 
k 4  4 mas
2

3

2
 dE 
C

 
4 m
 d (1/ as )  S

Adiabatic relation

C and 1/a are conjugate thermodynamic variables
1/a – „generalized force”
C - „generalize displacement” – capture physics at short length scales.
Shina Tan, Ann.Phys.323,2971(2008), Ann.Phys.323,2952(2008)
Other theory papers: Tan, Leggett, Braaten, Combescot, Baym, Blume, Werner, Castin, Randeria,Strinati,…

Unitary limit in 2 and 4 dimensions:
1

a   : R(r ) 
r

d 2

 O(r

4d

), Two body wave function for r  0.

Intuitive arguments:

R(r ) d r diverges at the origin

For d=2 the singularity of the wave function disapears = interaction
2

- For d=4
-

d

also disapears.

4D: noninteracting bosons
d 4

1

kF a

 

BCS

Strong
interaction

BEC

1

kF a

 

d 2

2D: noninteracting Fermi gas
The only nontrivial case of unitary regime is in 3D
Nussinov,Nussinov, Phys.Rev. A74, 053622(2006)

Hamiltonian

Hˆ  Tˆ  Vˆ   d 3r
Nˆ   d r
3



s 

2 



ˆ
 s (r )  
ˆ s (r )  g
 2m 

 nˆ (r )  nˆ (r )  ;

nˆs (r )  ˆ s (r )ˆ s (r )


3
d
 r nˆ (r )nˆ (r )

mkcut
1
m

 2 2
2
g
4 a 2

Path Integral Monte Carlo for fermions on 3D lattice

Volume  L3
L –limit for the spatial
correlations in the system

Coordinate space

lattice spacing  x
- Spin up fermion:

kcut



; x
x

- Spin down fermion:

External conditions:

T - temperature

 - chemical potential
Periodic boundary conditions imposed

Diagram. MC

QMC

Diagram. + analytic

Burovski et al.
PRL96, 160402(2006)

Bulgac, Drut, Magierski,
PRL99, 120401(2006)

Haussmann et al.
PRA75, 023610(2007)

Experiment

S. Nascimbene et al.
Nature 463, 1057 (2010)

exp( )

Courtesy of C. Salomon

Equation of state of the unitary Fermi gas - current status

Experiment: M.J.H. Ku, A.T. Sommer, L.W. Cheuk, M.W. Zwierlein , Science 335, 563 (2012)
QMC (PIMC + Hybrid Monte Carlo):
J.E.Drut, T.Lähde, G.Wlazłowski, P.Magierski, Phys. Rev. A 85, 051601 (2012)


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