magierski .pdf
File information
Original filename: magierski.pdf
Title: Slide 1
Author: Piotr Magierski
This PDF 1.5 document has been generated by MicrosoftÂ® PowerPointÂ® 2016, and has been sent on pdfarchive.com on 11/10/2017 at 14:47, from IP address 134.158.x.x.
The current document download page has been viewed 242 times.
File size: 3.7 MB (33 pages).
Privacy: public file
Share on social networks
Link to this file download page
Document preview
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 a3 >> 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, SchmittRink (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
4d
), 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)
Link to this page
Permanent link
Use the permanent link to the download page to share your document on Facebook, Twitter, LinkedIn, or directly with a contact by eMail, Messenger, Whatsapp, Line..
Short link
Use the short link to share your document on Twitter or by text message (SMS)
HTML Code
Copy the following HTML code to share your document on a Website or Blog