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−