DIAGRAMMATIC MONTE CARLO: From polarons to path-integrals (with worm, of course) Les Houches, June...
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Transcript of DIAGRAMMATIC MONTE CARLO: From polarons to path-integrals (with worm, of course) Les Houches, June...
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
����������������������������������������������������������������������������������������������������������������
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