DIAGRAMMATIC MONTE CARLO:

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DIAGRAMMATIC MONTE CARLO From polarons to
path-integrals (with worm, of course)
Nikolay Prokofiev, Umass, Amherst
Boris Svistunov, Umass, Amherst
Many thanks to collaborators on major algorithm
developments
Igor Tupitsyn, PITP
Vladimir Kashurnikov, MEPI, Moscow
Evgeni Burovski, Umass, Amherst
Andrei Mishchenko, AIST, Tsukuba
NASA
Les Houches, June 2006, Lecture 2
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Let
Diagram order
Contribution to the answer or the diagram
weight (positive definite, please)
Same-order diagrams
Integration variables
ENTER
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Polaron problem
electron
phonons
el.-ph. interaction
Green function


Sum of all Feynman diagrams
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Feynman digrams
Positive definite in momentum-imaginary
time representation
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Diagrams for
there are also diagrams for optical
conductivity, etc.
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Monte Carlo (Metropolis) cycle
Accept with probability
Diagram
suggest a change
Same order diagrams
Business as usual
Updating the diagram order
Ooops
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Balance Equation
If the desired probability density
distribution of diagrams in the stochastic sum
is (in most cases it is the same as the diagram
weight ) then the MC process of updating
diagrams should be stationary with respect to
(equilibrium condition)
Flux to
Flux out of
Is the probability density of making new
variables, if any
Detailed Balance solve it for each pair of
updates separately.
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Equation
Solution
Example
e.g.
for Frohlich polaron
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Lattice path-integrals for bosons and spins are
diagrams of closed loops!


imaginary time
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Diagrams for
Diagrams for
imaginary time
imaginary time
lattice site
lattice site
The rest is conventional worm algorithm in
continuous time
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I
I
M
M
I
I
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Path-integrals in continuous space are
diagrams of closed loops too!
P
2
1
P
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Not necessarily for closed loops!
Feynman (space-time) diagrams for fermions with
contact interaction (attractive)
Pair correlation function
connect vortexes with and
sum over all possible connections
Rubtsov 03 Burovski et al. 03
NOT EASY BUTTON
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NOT EASY BUTTON