nikolay prokofiev, umass, amherst

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Nikolay Prokofiev, Umass, Amherst work done in collaboration with PITP, Dec. 4, 2009 DIAGRAMMATIC MONTE CARLO FOR CORRELATED FERMIONS oris Svistunov UMass Kris van Houcke UMass Univ. Gent Evgeny Kozik ETH Lode Pollet Harvard Emanuel Gull Columbia Matthias Troyer ETH Felix Werner UMass

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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 Presentation

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Page 1: Nikolay Prokofiev, Umass, Amherst

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

Page 2: Nikolay Prokofiev, Umass, Amherst

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

Page 3: Nikolay Prokofiev, Umass, Amherst

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:

Page 4: Nikolay Prokofiev, Umass, Amherst

+

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)

Page 5: Nikolay Prokofiev, Umass, Amherst

{ , , }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

Page 6: Nikolay Prokofiev, Umass, Amherst

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

Page 7: Nikolay Prokofiev, Umass, Amherst

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”

Page 8: Nikolay Prokofiev, Umass, Amherst

-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!

Page 9: Nikolay Prokofiev, Umass, Amherst

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

Page 10: Nikolay Prokofiev, Umass, Amherst

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

Page 11: Nikolay Prokofiev, Umass, Amherst

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

Page 12: Nikolay Prokofiev, Umass, Amherst

“Exact” solution:

Polaron

Molecule

sure, press Enter

Updates:

Page 13: Nikolay Prokofiev, Umass, Amherst

/ 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

Page 14: Nikolay Prokofiev, Umass, Amherst

/ 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)

!

Page 15: Nikolay Prokofiev, Umass, Amherst

/ 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

Page 16: Nikolay Prokofiev, Umass, Amherst

/ 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

Page 17: Nikolay Prokofiev, Umass, Amherst

( )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

Page 18: Nikolay Prokofiev, Umass, Amherst

S. Nascimb`ene, N. Navon, K. J. Jiang, F. Chevy, and C. Salomon

DiagMC

ENS data

/Te

0/P P

Seatlle’s Det. MC

Page 19: Nikolay Prokofiev, Umass, Amherst

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

Page 20: Nikolay Prokofiev, Umass, Amherst

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

Page 21: Nikolay Prokofiev, Umass, Amherst

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

Page 22: Nikolay Prokofiev, Umass, Amherst

† † †' ' '

, , '

( ) 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)

Page 23: Nikolay Prokofiev, Umass, Amherst

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 …

Page 24: Nikolay Prokofiev, Umass, Amherst

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

Page 25: Nikolay Prokofiev, Umass, Amherst

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: