WORM ALGORITHM FOR CLASSICAL AND QUANTUM

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WORM ALGORITHM FOR CLASSICAL AND QUANTUM
STATISTICAL MODELS
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
Massimo Boninsegni, UAlberta, Edmonton
NASA
Les Houches, June 2006
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Worm algorithm idea
Consider
- configuration space arbitrary closed loops
- each cnf. has a weight factor
- quantity of interest
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conventional sampling scheme
local shape change
Add/delete small loops
No sampling of topological classes
can not evolve to
dynamical critical exponent in many
cases
Critical slowing down
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Worm algorithm idea
draw and erase
Masha
Ira
Ira

Masha
Masha
Masha
keep drawing
or
Topological classes are (whatever you can draw!)
No critical slowing down in most cases
Disconnected loops relate to important physics
(correlation functions) and are not merely an
algorithm trick!
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High-T expansion for the Ising model
where
3
4
2
1
4
4
2
number of lines enter/exit rule
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Spin-spin correlation function
Worm algorithm cnf. space
Same as for generalized partition
1
I
4
3
M
4
2
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Getting more practical since
Complete algorithm
- If , select a new site for
at random
- select direction to move , let it be bond

- If accept
with prob.
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IM
I
M
M
M
M
Correlation function
Magnetization fluctuations
Energy either
or
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Ising lattice field theory
expand
if
closed oriented loops
where
tabulated numbers
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Flux in Flux out closed oriented loops
of integer
N-currents
I
(one open loop)
M
Z-configurations have
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Same algorithm
IM
sectors, prob. to accept
M
M
draw
M
M
erase
Keep drawing/erasing
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Multi-component gauge field-theory (deconfined
criticality, XY-VBS and Neel-VBS quantum phase
transitions
XY-VBS transition understood (?) no DCP, always
first-order
Neel-VBS transition, unknown !
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Winding numbers
Homogeneous gauge in x-direction
Ceperley Pollock 86