nikolay prokofiev, umass, amherst
DESCRIPTION
DIAGRAMMATIC MONTE CARLO FOR CORRELATED FERMIONS. Nikolay Prokofiev, Umass, Amherst. work done in collaboration with . Lode Pollet Harvard. Matthias Troyer ETH. Evgeny Kozik ETH. Emanuel Gull Columbia. Boris Svistunov UMass. Kris van Houcke UMass Univ. Gent. Felix Werner - PowerPoint PPT PresentationTRANSCRIPT
Nikolay Prokofiev, Umass, Amherst
work done in collaboration with
PITP, Dec. 4, 2009
DIAGRAMMATIC MONTE CARLO FOR CORRELATED FERMIONS
Boris SvistunovUMass
Kris van HouckeUMass
Univ. Gent
Evgeny Kozik ETH
Lode PolletHarvard
Emanuel GullColumbia
Matthias TroyerETH
Felix WernerUMass
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
Fermi-Hubbard model: †
, ,
i j ii iij i i
H t a a U n n n
t U
† † †' ' '
, , '
( ) k k k q k q p q p kk kpq
H a a U a a a a
momentum representation:
Elements of the diagrammatic expansion:
THeaaG /
1,2,12)0(
,0)()(Tr),( kkk
)( 21 qU
1
2
qk
qp p
q
k
k1 21 2( )qU
(0)3 4( , )G k
(0)
5 6( , )G p
fifth order term:
+
The full Green’s Function:
TH
pp eaapG /1,2,12, )()(Tr),(
+ +
+ …+ + +
+0 ( , )G p
=0 ( , )G p
Why not sample the diagrams by Monte Carlo?
Configuration space = (diagram order, topology and types of lines, internal variables)
{ , , }i i iq p
Diagram order
Diagram topology
MC update
MC updateM
C up
date
This is NOT: write diagram after diagram, compute its value, sum
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 n Cluster DMFT / DCA methods+ universal- cluster size extrapolation
Diagrammatic MC+ universal- diagram-order extrapolation
Cluster DMFT
linear size
N diagram order
Diagrammatic MC
DF LT
Computational complexityIs exponential :exp{# }
for irreducible diagrams
Further advantages of the diagrammatic technique
Calculate irreducible diagrams for , , … to get , , …. from Dyson equations
+ + + ...0 ( , )G p
1 2( , )p
G U
+ Dyson Equation:( , )G p
Make the entire scheme self-consistent, i.e. all internal lines in , , … are “bold” = skeleton graphs
U +U
or(0)
+ U (0)G
+ + T
G(0) (0)GG G G
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: environmentparticl couplinge HHH H
quasiparticle
*( 0), , ( , ), ...E p m G p t
E.g. 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
electron
phonons
el.-ph. interaction
Fermi-polaron problem:
Fermi s
2
ea ( )' ( ') ' 2
H V rH nr r rpm
d
0rr
( )V r
( )r
1/3 ~ /Fn k
0
~ 10
F S
F
kk ra
Universal physics( independent)( )V r
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.
G
G
Sign problem! max 11N
self-consistent and G
self-consistent onlyG
=( )Fk a
0.615Confirmed
by ENS
“Exact” solution:
Polaron
Molecule
sure, press Enter
Updates:
/ 4U t / 1.5 0.6t n
/ 0.025 /100FT t E
2D Fermi-Hubbard model in the Fermi-liquid regime
Bare series convergence:yes, after order 4
Fermi –liquid regime was reached
2'
'
2
2 2
( ) (0) ( )6
( ) (0)6
F FF
F
TE T E E
Tn T n
/ 4U t / 3.1 1.2t n
/ 0.4 /10FT t E
2D Fermi-Hubbard model in the Fermi-liquid regime
Comparing DiagMC with cluster DMFT(DCA implementation)
!
/ 4U t / 3.1 1.2t n
/ 0.4 /10FT t E
2D Fermi-Hubbard model in the Fermi-liquid regime
Momentum dependence of self-energy
0( , ) x yT p p p along
/ 4U t ( ) / 1.5 1.35nU t n
/ 0.1 / 50FT t E
3D Fermi-Hubbard model in the Fermi-liquid regime
DiagMC vs high-T expansion in t/T(up to 10-th order)
Unbiased high-T expansion in t/Tfails at T/t>1 before the FL regime sets in
28
10
10
8
( )Fk a / 0.3 / 0.152(7)F
C F
T ET E
3D Resonant Fermi gas at unitarity :
Bridging the gap between different limits
S. Nascimb`ene, N. Navon, K. J. Jiang, F. Chevy, and C. Salomon
DiagMC
ENS data
/Te
0/P P
Seatlle’s Det. MC
700
600
500
400
300
200
100
1000900800700600500
2.5
2.0
1.5
1.0
0.5
0.0
Opt
ical
Den
sity
300250200150100500
Pixel [1.2 m]
DiagMC fit
Andre Schirotzek, Ariel Sommer, Mark Ku, and Martin Zwierlein
60
50
40
30
20
10
0
F 0
3210-1-2-3
Universal function F0
Single shot data
3 Tn
Andre Schirotzek, Ariel Sommer, Mark Ku, and Martin Zwierlein
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 ( version) and the continuous electron gas or jellium model (screening version). • Next step: Effects of disorder, broken symmetry phases, additional correlation functions, etc.
G
† † †' ' '
, , '
( ) k k k q k q p q p kk kpq
H a a U a a a a
1 2( )qU
(0)3 4( , )G k
(0)
5 6( , )G p
Configuration space = (diagram order, topology and types of lines, internal variables)
1 2 1 20
; , , ,n nnn
A y d x d x d x W x x x y W
term order
different terms ofof the same order
Integration variables
Contribution to the answer
Monte Carlo (Metropolis-Rosenbluth-Teller) cycle:
Diagram suggest a change
Accept with probability
'
'
1(new { })acc
v v
WR
W P x
Collect statistics: ( )counter counterA y A y sign
sign problem and potential trouble!, but …
Fermi-Hubbard model:
( , )G p
( )U q
( , )G p
Self-consistency in the form of Dyson, RPA
+ G U
+ U
Extrapolate to the limit.N
(0)G
†
, ,
i j ii iij i i
H t a a U n n n
Large classical system Large quantum system
#Nstate described by numbers # Nestate described by numbers
Possible to simulate “as is” (e.g. molecular dynamics)
Impossible to simulate “as is”
approximate Math. mapping forquantum statistical predictions
Feynman Diagrams: