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 rp

md

0rr

( )V r

( )r

1/3 ~ /Fn k

0

~ 1

0F S

F

k

k r

a

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 E

T 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

F0

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 2

0

; , , ,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 { })accv 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: