DIAGRAMMATIC MONTE CARLO:

From polarons to path-integrals (with worm, of course)

Les Houches, June 2006, Lecture 2

Nikolay Prokofiev, Umass, Amherst

Boris Svistunov, Umass, Amherst

Igor Tupitsyn, PITP

Vladimir Kashurnikov, MEPI, Moscow

Evgeni Burovski, Umass, Amherst

Andrei Mishchenko, AIST, Tsukuba

Many thanks to collaboratorson major algorithm developments

NASA

Let …

1 2 1 2

0

; , , ,n nnn

A y d x d x d x D x x x y

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Diagram order

Same-order diagrams

Integration variables

Contribution to the answer or the diagram weight(positive definite, please)

ENTER

Polaron problem: ( ) ( )( 1/ 2) . .p p q q q p q p qp q pq

H p a a p b b V a a b h c

electron phonons el.-ph. interaction

Green function: ( , ) (0) ( )p pG p a a ( , )G p

p

0 + 0 1 2 p q

pp

q

+ …

Sum of all Feynman diagrams

( , )G p Feynman digrams

1 4 80

1q2q

3q4q

p1p q

21k 2 1( )( )ke

qV

2 1( )( )qe 1 2

q

Positive definite inmomentum-imaginary time

representation

1 2 1 2

0

; , , ,n nnn

A y d x d x d x D x x x y

����������������������������������������������������������������������������������������������������������������

Diagrams for: (0) (0) (0) ( ) ( ) ( )B A A B A B A Bq q p q q p q q q qb b a a b b

there are also diagrams for optical conductivity, etc.

1 2 1 2

0

; , , ,n nnn

A y d x d x d x D x x x y D

����������������������������������������������������������������������������������������������������������������

' ( )( )

( )

n mm

n

D d xd x

D d x

Monte Carlo (Metropolis) cycle:

Diagram D suggest a change

Accept with probability

Same order diagrams:Business as usual

Updating the diagram order:

' ( )(1)

( )

n

n

D d xO

D d x

'acc

DR

D

Ooops

Balance Equation: If the desired probability density distributionof diagrams in the stochastic sum is(in most cases it is the same as the diagram weight )then the MC process of updating diagrams shouldbe stationary with respect to (equilibrium condition)

D

P

' '' '

' '

( ') ( )accept acceptupdates updates

P W R P W R

Flux out of Flux to

( ')W Is the probability density of “making” new variables, if any

P

Detailed Balance: solve it for each pair of updates separately.

' '' '( ') ( )accept acceptP W R P W R

Equation:

11 1 ,, , ( ) , , ( ) , , ( )acceptn

am n n m n m n m n

n n n m n mn n m n m cceptD x x d x W x x d x R D x x d x g R

e.g.

D 'D

Solution:1

1 1

( , , )

( , , ) ( , , )

n n maccept n m n m n mn m naccept n n n n m

R D x x gR

R D x x W x x

20 2 1 2 1( ) ( / 2 )( )1 q mW e e

Example:

for Frohlich polaron

20/ 2 , , /q qp m V q

1/( )g n m

0

Lattice path-integrals for bosons and spins are “diagrams” of closed loops!

10 ( , )ij i j i iij

i j j iiji

H t n n b bH H U n n n

1

0 0

0

( )

1 1 1

0 0

Tr Tr

Tr 1 ( ) ( ) ( ) ' ...

H dHH

H

Z e e e

e H d H H d d

i j

imag

inar

y ti

me

0

+

0,1,2,0in i j

'+

(1, 2)t

Diagrams for im

agin

ary

tim

e

lattice site

-Z= Tr e H

Diagrams for

† -M= Tr T ( ) ( ) eI M IM

HIb bG

0

imag

inar

y ti

me

lattice site

0

I

M

The rest is conventional worm algorithm in continuous time

M

I I

II

M

Path-integrals in continuous space are “diagrams” of closed loops too!

2/1

11

( )... exp ( )

2

Pi i

Pi

m R RZ dR dR U R

P

1

2

P

1 2, , ,( , , ... , )i i i i NR r r r 1,ir 2,ir

Not necessarily for closed loops!

Feynman (space-time) diagrams for fermions with contact interaction (attractive)

1 1 1 1 2 2 2 2( , ) ( , ) ( , ) ( , )a r a r a r a r

Rubtsov ’03Burovski et al. ’03

U

connect vortexes with and GG

perm(( ) )( ) 1n nn G G GD U drdG

2( ) ( ) ( , )t ( )de 0n ni jnD x xG drdU

sum over all possible connections2( !)n

NOT EASY BUTTON

Pair correlation function

2( ) ( ) ( , )t ( )de 0n ni jnD U x xG drd

NOT EASY BUTTON