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DIAGRAMMATIC MONTE CARLO FOR CORRELATED FERMIONS
Nikolay Prokofiev, Umass, Amherst
work done in collaboration with
Boris Svistunov Kris van Houcke Felix Werner
UMass
UMass
UMass
Univ. Gent
Evgeny Kozik Lode Pollet Emanuel Gull Matthias Troyer
Harvard
ETH
ETH
Columbia
PITP, Dec. 4, 2009
Outline
Feynman Diagrams: An acceptable solution to the sign problem?
(i) proven case of fermi-polarons
Many-body implementation for
(ii) the Fermi-Hubbard model in the Fermi liquid regime and
(iii) the resonant Fermi gas
H  t
Fermi-Hubbard model:

ij  ,
t
ai a †j  U  ni ni    ni
i ,
i
U
momentum representation:
H   ( k   )ak† ak 
k ,

kpq , '
U q ak† q a †p  q ' a p ' ak '
Elements of the diagrammatic expansion:
fifth order term:
G (k ,  2   1 )  Tr a,k ( 2 ) a,k ( 1 ) e

( 0)
 ,
U q ( 1   2k)
H0 /T


G(0) (k , 3  4 )
1
k
U q  ( 1   2 )
1
2
k q
G(0) ( p, 5  6 ) q
p
2
pq
G, ( p,  2   1 )  Tr a, p ( 2 ) a, p ( 1 ) e  H / T 


The full Green’s Function:
G0 ( p, ) =
+
G0 ( p, )
+
+
+
+
+
+
+…
Why not sample the
diagrams by Monte Carlo?
Configuration space = (diagram order, topology and types of lines, internal variables)
Diagram order
{qi , i , pi }
Diagram topology
This is NOT: write diagram after diagram, compute its value, sum
Sign-problem
Variational methods
Determinant MC
+ “solves” ni  ni   1 case
+ universal
- often reliable only at T=0
- systematic errors
- finite-size extrapolation
- CPU expensive
- not universal
- finite-size extrapolation
Cluster
DMFT /MC
DCA methods
Diagrammatic
+ universal
- diagram-order
cluster size extrapolation
extrapolation
Computational complexity
Is exponential : exp{#  }
Cluster DMFT
 F
T
 
 D
L

linear size
Diagrammatic MC
N
diagram order
for irreducible
diagrams
Further advantages of the diagrammatic technique
Calculate irreducible diagrams for  ,  , … to get G , U , …. from Dyson equations

G ( p, )
+
Dyson Equation:
G0 ( p, )


+
(
p,1   2 ) +


 (0)
+
+ ...

or
U
U
+
+
G (0)
U
+
GG  G (0)G (0)
T
G
Make the entire scheme self-consistent, i.e. all internal lines in  ,  , … are “bold”
= skeleton graphs
Every analytic solution or insight into the problem can be “built in”
-Series expansion in U is often divergent, or, even worse, asymptotic.
Does it makes sense to have more terms calculated?
Yes! (i) Unbiased resummation techniques
(ii) there are interesting cases with convergent series
(Hubbard model at finite-T, resonant fermions)
- It is an unsolved problem whether skeleton diagrams form
asymptotic or convergent series
Good news: BCS theory is non-analytic at U 0 ,
and yet this is accounted for within
the lowest-order diagrams!
Polaron problem:
H  H particle  H environment  H coupling 
quasiparticle
E ( p  0), m* , G( p, t ), ...
E.g. Electrons in semiconducting crystals (electron-phonon polarons)
H    ( p)a p a p 
electron
p
e
e
  ( p)(b b

q q
 1/ 2) 
phonons
q
 V a
q
pq

p q
a p bq  h.c.
el.-ph.
interaction
2
p
Fermi-polaron problem: H 
 H Fermi sea   V (r  r ') n(r ') dr '
2m
V (r )
 (r )
k F aS ~ 1
r0
r
k F r0  0
Universal physics
( V ( r ) independent)
n1/ 3 ~ k F / 
Examples:
Electron-phonon polarons (e.g. Frohlich model)
= particle in the bosonic environment.
Too “simple”, no sign problem, N
10 2
G
Fermi –polarons (polarized resonant Fermi gas
= particle in the fermionic environment.
G

self-consistent G and 
( k F a )
=
self-consistent G only
0.615
Confirmed
by ENS
Sign problem!
N max  11
“Exact” solution:


Polaron


Molecule
sure, press
Updates:
Enter
2D Fermi-Hubbard model in the Fermi-liquid regime
U /t  4
 / t  1.5  n  0.6
T / t  0.025 EF /100
Fermi –liquid regime was reached
 T
Bare series convergence:
E (T )  E (0)  (  F   F' EF )
yes, after order 4 6
2
n(T )  n(0)   F'
 2T 2
6
2
2D Fermi-Hubbard model in the Fermi-liquid regime
U /t  4
 / t  3.1  n  1.2
T / t  0.4 EF /10
Comparing DiagMC with cluster DMFT
(DCA implementation)
!
2D Fermi-Hubbard model in the Fermi-liquid regime
U /t  4
 / t  3.1  n  1.2
T / t  0.4 EF /10
Momentum dependence of
self-energy
(0   T , p) along px  p y
U /t  4
(   nU ) / t  1.5  n  1.35
3D Fermi-Hubbard model in the Fermi-liquid regime
T / t  0.1 EF / 50
DiagMC vs high-T expansion in t/T
(up to 10-th order)
10
8
8
Unbiased high-T expansion in t/T
fails at T/t>1 before the FL regime sets in
2
10
3D Resonant Fermi gas at unitarity (k F a  ) :
Bridging the gap between
different limits
T / EF  0.3
TC / EF  0.152(7)
S. Nascimb`ene, N. Navon, K. J. Jiang, F. Chevy, and C. Salomon
ENS data
DiagMC
P / P0
Seatlle’s Det. MC
e   /T
Andre Schirotzek, Ariel Sommer, Mark Ku, and Martin Zwierlein
0
0
2.5
2.0
Optical Density
0
0
0
1.5
DiagMC fit
1.0
0.5
0.0
0
0
50
100
150
Pixel [1.2 m]
0
500
600
700
800
900
1000
200
250
300
Andre Schirotzek, Ariel
Sommer,
MarkF Ku, and Martin Zwierlein
Universal
function
0
Single shot data
60
50
3
T
F0
n
40
30
20
10
0
-3
-2
-1
0


1
2
3
Conclusions/perspectives
• Bold-line
Diagrammatic series can be efficiently simulated.
- combine analytic and numeric tools
- thermodynamic-limit results
- sign-problem tolerant (small configuration space)
• Work
in progress: bold-line implementation for the Hubbard
model and the resonant Fermi-gas (
G
version) and the continuous
electron gas or jellium model (screening version).
• Next
step: Effects of disorder, broken symmetry phases, additional
correlation functions, etc.
H   ( k   )a ak 
k ,
†
k
 U a
q
kpq ,
†
k  q
a
†
p  q '
a p ' ak '
'
U q ( 1   2 )
G(0) (k , 3  4 )
G(0) ( p, 5  6 )
Configuration space = (diagram order, topology and types of lines, internal variables)


A y    d x1d x 2
n 0 

d x n Wn  ; x1 , x 2 ,

x n , y  W

Integration variables
term order
Contribution to the answer
different terms of
of the same order
Monte Carlo (Metropolis-Rosenbluth-Teller) cycle:
Accept with probability
Diagram

Collect statistics:
suggest a change
 
Racc 
W '
1

W Pv v ' (new {x})
 
Acounter y  Acounter y  sign( )
sign problem and potential trouble!, but …
Fermi-Hubbard model:
H  t

ij  ,
ai a †j  U  ni ni    ni
i ,
i
G ( p, )
U (q)
G ( p, )
Self-consistency in the form of Dyson, RPA
G

G
(0)
Extrapolate to the
+


N   limit.
U
U
+

Large classical system
state described by
#N
numbers
Large quantum system
state described by
Possible to simulate “as is”
(e.g. molecular dynamics)
e# N
numbers
Impossible to simulate “as is”
approximate
Math. mapping for
quantum statistical predictions
Feynman Diagrams: