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WORM ALGORITHM
Nikolay Prokofiev, Umass, Amherst
Boris Svistunov, Umass, Amherst
Igor Tupitsyn, PITP, Vancouver
Vladimir Kashurnikov, MEPI, Moscow
Massimo Boninsegni, UAlberta, Edmonton
NASA
ISSP, August 2006
Evgeni Burovski, Umass, Amherst
Matthias Troyer, ETH
Masha
Ira
Why bother with algorithms?
PhD while still young
( , )G r τ
New quantities, more theoreticaltools to address physics
Grand canonical ensembleOff-diagonal correlations“Single-particle” and/or condensate wave functionsWinding numbers and
Examples from : superfluid-insulator transition, spin chains, helium solid & glass, deconfined criticality,resonant fermions, holes in the t-J model, …
S ρ ( )rϕ
( )N µ
Efficiency
PhD while still youngBetter accuracyLarge system sizeMore complex systemsFinite-size scalingCritical phenomena Phase diagrams
Reliably!
Worm algorithm idea
Consider:
- configuration space = arbitrary closed loops
- each cnf. has a weight factor c n fW
- quantity of interest c n fAc n f c n f
c n f
c n fc n f
A W
AW
=∑∑
“conventional”sampling scheme:
local shape change Add/delete small loops
can not evolve to
No sampling of topological classes(non-ergodic)
Critical slowing down(large loops are related tocritical modes)
updates zd
NL
L
∼
dynamical critical exponent in many cases2z ≈
Worm algorithm idea
draw and erase:
Masha
Ira
or
Masha
Ira+
keepdrawing
Topological are sampled (whatever you can draw!)
No critical slowing down in most cases
Disconnected loops are related to correlation functions and are not merely an algorithm trick!
Masha
Masha
High-T expansion for the Ising model ( 1)i jij
HK
T < >
− = σ σ σ = ±∑
/
{ } { } { }
cosh (1 tanh )i i
i j
i
H T
b ij bi j
i
K
j
e KZ e K−
σ σ σ=< > =<
σ
>
σ + σ σ= ≡ ≡∑ ∑ ∑∏ ∏
( ){ } 0,1 { 0,1} 1
~ tanh tanh i
i
bb
i b b
N Ni j
N Nb ij b ij
Mi
i
K Kσ
σ σ= = σ = ±=< > =< >
σ
= ∑ ∑ ∑∑∏ ∏ ∏
i b ijij
M N even=< >< >
= =∑{ }
~ tanhb
b
N loops
N
b ij
K=< >=
∑ ∏
Graphically:
number of lines;continuity (enter/exit)
bN =iM even→ =
2 4
2
/
{ }i
H TI MM IG e−
σσ σ= ∑
{ } 1
tanh b i
b
i
i
I iMN Mi
N b ij i
G K +δ +δ
σ =±=< >
≡ σ ∑ ∑∏ ∏
Spin-spin correlation function: ,IMIM
Gg
Z=
{ }
~ tanhb
b
N loopsIM w
N
b ijorm
K=<= >
+
∑ ∏
2
41I
M
3
Worm algorithm cnf. space = Z G�WZ Z G= + κ
Same as for generalized partition
Complete algorithm:
- If , select a new site for at random
- select direction to move , let it be bond
- If accept wit h prob.
I M=
M b
,I M
R =1
min(1, tanh( ))
min(1, tanh ( ))
K
K−bN = 0
1bN → 1
0
Easier to implement then single-flip!
I=M
M
I
M
M
M
,
( ) ( ) 1
I M
bonds bonds bb
G I M G I M
Z Z
N N N
− = − +
= + δ = + ∑
Correlation function: ( ) ( ) /g i G i Z=
Energy: either 1 2 (1)E JNd JNdg= − σ σ = −
Magnetization fluctuations: ( )22 ( )iM N g i= σ = ∑ ∑
or 2tanh( ) sinh ( )bondsE J K dN N K = − +
MC estimators
Ising lattice field theory( , ,
2 4
, )x y zi i i i
i i i
Ht U
T∗
+ν=±
ν
− = ψ ψ + µ ψ − ψ ∑ ∑ ∑
/H
i
Ti eZ d −= ψ ∏ ∫
expand0
( )
!i i
N Nt i i
N
te
N
∗+ν
∗∞ψ ψ +ν
=
ψ ψ= ∑
( )( )2 42
1
{ } , !
ii i ii
i
NM UM
i i iN i ii
tZ d e
N
ν
ν
µ ψ − ψ∗
ν ν
= ψ ψ ψ ∑ ∏ ∏ ∫
1 21 2( )i M M
M M Me ϕ − = =→
( )ii
Q M∏
where ( )Q M =2
1 2
0
0
M x Ux
M M
dx x eπ∞
µ −
≠
=∫if closed oriented loops
tabulated numbers
i i + ν
'i + ν
iN ν
', 'iN +ν −ν
,( )i i
i i
N N ν +ν −νν ν∗ψ ψ∑ ∑
Flux in = Flux out closed oriented loopsof integer N-currents
( )( ) I M
G I Mg I M
Z∗−− = = ψ ψ
(one open loop)
M
I
Worm algorithm cnf. space = Z G�
Same algorithm:
sectors, prob. to acceptZ G↔
I=M
draw1M MN N ν ν→ + '
'
( 1)min 1,
( 1) ( )M
M M
t Q MR
N Q M ν
+= + , , 1M MN N+ν −ν +ν −ν→ −
( 1)min 1,
( )I
z GI
Q MR
Q M→
+= ,( ) ( 1)
min 1,( )
M M
M
N Q MR
t Q M+ν −ν −
= erase
M
M
M
M
Keep drawing/erasing …
Multi-component gauge field-theory(deconfined criticality, XY-VBS and Neel-VBS quantum phase transitions…
( ) 2 2 2
, , , , ,2
; ; ;
[ ( )]a i a i a i ab a i b ia i a i i
i
ab
A ie A iH
t UT
ν∗+ν �ν
ν− = ψ ψ + µ ψ − ψ ψ − κ ∇× ∑ ∑ ∑∑
1A+2A+
3A−4A−
11 22 12U U U= ≠
11 22 12U U U= =
XY-VBS transitionno DCP, always first-order
Neel-VBS transition, unknown !
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 +
< >
+ µ −= = − ∑ ∑ ∑
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
β
− τ τ−β−β
β β β−β
τ
∫= ≡ = − τ τ + τ τ τ τ + ∫ ∫ ∫
i j
imag
inar
y tim
e
0
β
+
0,1,2,0in =i j
τ'τ
+
(1,2)t
Diagrams forim
agin
ary
time
lattice site
-Z=Tr e Hβ
Diagrams for
† -M= Tr T ( ) ( )eI M IM
HIb bG β
τ τ τ
0
β
imag
inar
y tim
e
lattice site
0
β
I
M
The rest is conventional worm algorithm in continuo us time
(there is no problem to work with arbitrary number of continuousvariables as long as an expansion is well defined)
( ) ( )1 2 1 2
0
; , , ,n nnn
A y d x d x d x D x x x yξ
ξ∞
=
=∑∑∫∫∫ur r r r r r r ur
K K
Diagram order
Same-order diagrams
Integration variables
Contribution to the answeror the diagram weight
(positive definite, please)
ENTER
Diagrammatic Monte Carlo (not in this lecture)
M
II
II
M
Path-integrals in continuous space are consist of closed loops too!
2/1
11
( )... exp ( )
2
Pi i
Pi
m R RZ dR dR U R
=β τ +
=
− = − + τ τ ∑∫∫∫
P
1
2
P
1 2, , ,( , , ... , )i i i i NR r r r = 1,ir 2,ir P
βτ =
2
( )2
ii j
i i j
pH V r r
m <= + −∑ ∑
Feynman path-integral
( , )r tψ
( ', ')r tψ +
ZG
diagrammatic expansionfor ( ) 0V r <
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 rτ τ τ τ+ +↑ ↓ ↓ ↑
U= −
connect vortexes with and G↑G↓
perm(( ) )( ) 1n nn G G GD U drdG τ↑ ↑ ↓ ↓ −= −
rK K2( ) ( ) ( , )t ( )de 0n n
i jnD x xG drdUξ
ξ τ↑= − ≥∑ r r rsum over all possible connections2( !)n
Pair correlation function
,
( ) ( )i i ji i j
H k V r rσ ↑ ↓σ=↑↓ <
= ε + −∑ ∑
Worm cnf. space = ( )Z G =・ ? ?disconnected loop
Domain walls in 2D Ising model are loops!
G space is NOT necessarily physical
Disconnected loop is unphysical!
Ira
Masha
?
?
Quantum spin chainsmagnetization curves, gaps, spin wave spectra
[ ( ) ]H x jx ix jy iy z jz iz izij i
J S S S S J S S H S< >
= − + + − ∑ ∑
S=1/2 Heisenberg chain
Bethe ansatz (M. Takahashi)
MC data
Line is for the effective fermion
theory with spectrum
2 2( 2 / )p n L cpπε = = ∆ + 0.4105(1)∆ =
2.48(1)c =
Lou, Qin, Ng, Su, Affleck ‘99deviations are due to magnon-magnon interactions
( ) '
†
2( )†
1''
= T (x, ) (0)( , )
G
ipxI
E Ep JG
G e dx S S
S e Z e
p τ
τ ττ
τ
α
−
− −∆>>α
α
=
τ
→
∫∑
One dimensional S=1 chain with / 0.43z xJ J =
0.02486(5)∆ =Spin gap
Z -factor 0.980(5)Z =
Energy gaps:
( )
( , 1)1
e eG p Z
e
−∆τ −∆ β−τ
−∆β
++
τ >> ≈
Spin waves spectrum: One dimensional S=1 Heisenberg chain
Worm algorithm = cnf. space of + updates based on Z G� Masha
Ira
Conclusions:
Can be formulated for:
- classical and quantum models (Bose/Fermi/Spin)
- different representations (path-integrals, Feynam di agrams,SSE)
- non-local solutions of balance Eqn. (clusters, dire cted loops, geom. worm)
Winding numbers
222
2 2
xdS d
T WFL
L−
−
∂Λ = =∂ϕ
CeperleyPollock ‘86
Homogeneous gauge in x-direction: , ( ) / , ( )iAx xt te A r L A r dx → = ϕ = ϕ∫
( )2/
ln (0)2
x
x
x
i WW
W
dS
Z e Z
LF T Z F L
ϕ=
ϕ= − = + Λ
∑, ,( ) / 1x i x i x x
i
W N N L + = − =∑
iAe
iAe
iAe−
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