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Nikolay Prokofiev, Umass, Amherst
Boris SvistunovUMass, Amherst
work done in collaboration with
QS 2009, IPAM
DIAGRAMMATIC MONTE CARLO: WHAT HAPPENS TO THE SIGN‐PROBLEM
Kris van HouckeUMass, Amherst,
Univ. Gent
Evgeny KozikUMass, Amherst,
ETH, Zurich
Outline
Ising spins vs Feynman Diagrams: Is there any difference from the Monte Carlo perspective?
Polarons in Fermi systems
Many‐body implementation for the Fermi‐Hubbard model
Acceptable solution to the sign problem?
Yes! (so far …)
Feynman Diagrams: graphical representation for the high‐order perturbation theory
int0 H HH= +
/
0 0 0/ ...
H Tn n
nH T
n nn
AeA A AB AC
e
−
−
Ψ Ψ= = + + +
Ψ Ψ
∑∑
int H∝ 2int H∝
explicit graphical representation for all terms and easy rules to convert graphs to math
Diagrammatic technique: explicit summation of geometric series “on‐the‐go” with self‐consistent re‐formulation of the diagrams
† † †' ' '
, , '
( ) k k k q k q p q p kk kpq
H a a U a a a aσ σ σ σ σ σσ σσ
ε μ − += − +∑ ∑
1 2( )qU δ τ τ−
(0)3 4( , )G k τ τ
↓ −(0)
5 6( , )G p τ τ ↑ −
Configuration space = (diagram order, topology and types of lines, internal variables)
Standard Monte Carlo setup:
‐ each cnf. has a weight factor c n fW
‐ quantity of interest
c n f c n fc n f
c n fc n f
A WA
W=
∑∑
‐ configuration space (depends on the model and it’s representation)
/c n fE Te −
Monte Carlo
MC
cnfcnf
A ∑ configurations generated from the prob. distribution
cnfW
( ) ( )1 2 1 20
; , , ,n nnn
A y d x d x d x W x x x y Wνξ ν
ξ∞
=
= =∑∑ ∑∫∫∫ur r r r r r r ur
K K
term order
different terms ofof the same order
Integration variables
Contribution to the answer
Monte Carlo (Metropolis‐Rosenbluth‐Teller) cycle:
Diagram ν suggest a change
Accept with probability
'
'
1(new { })acc
v v
WR
W P xν
ν
→
∼ v
Collect statistics: ( ) ( ) ( )counter counterA y A y sign ν= +ur ur
sign problem and potential trouble!, but …
Sign‐problem
Variational methods+ universal‐ often reliable only at T=0‐ systematic errors‐ finite‐size extrapolation
Determinant MC+ “solves” case ‐ CPU expensive ‐ not universal‐ finite‐size extrapolation
1i in nσ σ−+ =Cluster DMFT / DCA methods+ universal‐ cluster size extrapolation
Diagrammatic MC+ universal‐ diagram‐order extrapolation
Cluster DMFT
linear size
Nξ =diagram order
Diagrammatic MC
DF LTεξ ⎛ ⎞= ⎜ ⎟
⎝ ⎠
Computational complexityIs exponential : exp{# }ξ
for irreducible diagrams
Further advantages of the diagrammatic technique
Calculate irreducible diagrams for , , … to get , , …. from Dyson equations
+ + + ...=0 ( , )G p τ
1 2( , )p τ τΣ −
Σ Π G U
+ =Dyson Equation:( , )G p τ
Make the entire scheme self‐consistent, i.e. all internal lines in , , … are “bold”Σ Π
U +=
UΠ
Σ
or(0) Γ
+ =U (0)G
Γ + + = T
G(0) (0)GG G G −
Polaron problem:
environmentparticl couplinge HHH H= + + → quasiparticle
*( 0), , ( , ), ...E p m G p t=
Electrons in semiconducting crystals (electron‐phonon polarons)
e e
( )
( )( 1/ 2)
( )
. .
q qq
q
p p
pqq
p
p q p
H p a
V
a
a a
p
h
b
c
b
b
ω
ε
+
+
+−
+
= +
++
+
∑
∑
∑
electron
phonons
el.‐ph. interaction
Fermi‐polaron problem:Fermi s
2
ea ( )' ( ') '2
H V rH nr r rpm
d= + + −∫
0rr
( )V r
( )rΨ
1/3 ~ /Fn k π
0
~ 10
F S
F
kk r
a→
Universal physics( independent)( )V r
particle dressed by interactions with the Fermi sea:
Non‐interacting Fermi sea
Extra fermion
Fermionic quasiparticle (polaron)
BCS BEC
Bosonic quasiparticle (molecule)
cold Fermi gases with population strong imbalance orthogonality catastrophe, X‐ray singularities, heavy fermions,
quantum diffusion in metals, ions in He‐3, etc.
Examples:
Electron‐phonon polarons (e.g. Frohlich model)= particle in the bosonic environment.
Too “simple”, no sign problem, 210N :
Fermi –polarons (polarized resonant Fermi gas= particle in the fermionic environment.
Γ
G↑
G↓
Sign problem! max 11N =self‐consistent and G Γ
self‐consistent onlyG
Fermi‐Hubbard model:
( , )G p τ↑
( )U q
( , )G p τ↓
Self‐consistency in the form of Dyson, RPA
+ = ΣG U
+ =U
Π
Extrapolate to the limit.N → ∞
(0)G
†
, , i j ii i
ij i iH t a a U n n nσ σ σ
σ σ
μ ↑ ↓< >
= − + −∑ ∑ ∑
Evgeni Kozik
Kris Van‐Houcke
1D
/ 4U t =/ 0.5tμ =−
/ 0.3T t =
3D
/ 4U t =/ 1.5tμ =
/ 0.5T t =
Better quality than in 1D!
In 3D temperatures are low enough to claim Fermi‐liquid properties / 0.03T zt <
(using bare propagators so far)
Conclusions/perspectives
• Bold-line Diagrammatic series can be efficiently simulated.- combine analytic and numeric tools
- thermodynamic-limit results
- sign-problem tolerant (relatively small configuration space)
• Work in progress: bold-line implementation for the Hubbard
model and the resonant Fermi-gas ( version) and the continuous
electron gas, or jellium model (screening version).
• Next step: Effects of disorder, broken symmetry phases, additional correlation functions, etc.
G Γ
Γ
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