worm algorithms - ustc

Outline Worm algorithms for Ising high-temperature graphs Worm algorithms for fully-packed loops Summary

Worm Algorithms

Youjin Deng

Department of Modern PhysicsUniversity of Science and Technology of China

P.R.China

November 6Hefei


Outline

References/Collaborators

1. Youjin Deng, Timothy M. Garoni, and Alan D. Sokal,Dynamic Critical Behavior of the Worm Algorithm for the Ising Model,Phys. Rev. Lett. 99, 110601 (2007).

2. Wei Zhang, Timothy M. Garoni, and Youjin Deng,Simulating the fully-packed loop model on the honeycomb lattice with aworm algorithm, preprint.


Outline

How do we efficiently simulate models near criticality?

◮ Problem: critical slowing-down


Outline


◮ Problem: critical slowing-down◮ The current state-of-the-art: cluster algorithms

◮ Swendsen & Wang PRL 1987◮ Use global moves in clever way


Outline


◮ Problem: critical slowing-down◮ The current state-of-the-art: cluster algorithms

◮ Swendsen & Wang PRL 1987◮ Use global moves in clever way

◮ More recent idea: worm algorithms◮ Prokof’ev & Svistunov PRL 2001◮ Enlarge an Eulerian configuration space to include defects◮ Move the defects via random walk


High-temperature expansions, state spaces, worm dynamics . . .

Eulerian subgraphs◮ Fix a finite graph G = (V , E)




◮ A ⊆ E is Eulerian if every vertex in (V , A) has even degree




◮ A ⊆ E is Eulerian if every vertex in (V , A) has even degree◮ The cycle space C(G) = {A ⊆ E : A is Eulerian}





◮ Consider the Ising model on G

ZIsing =∑

σ∈{−1,+1}V

∏

ij∈E

eβσiσj





◮ Consider the Ising model on G

ZIsing =∑

σ∈{−1,+1}V

∏

ij∈E

eβσiσj

◮ The high-temperature expansion is

ZIsing =(

2|V | cosh|E | β) ∑

A∈C(G)

(tanh β)|A|



State space for worm dynamics

◮ Let ∂A be the set of all vertices with odd degree in (V , A)

x

y◮ For distinct x , y ∈ V define

Sx ,y = {A ⊆ E : ∂A = {x , y}}

Sx ,x = {A ⊆ E : ∂A = ∅}

◮ In this notation Sx ,x = C(G)





x


Sx ,y = {A ⊆ E : ∂A = {x , y}}

Sx ,x = {A ⊆ E : ∂A = ∅}


◮ State space of worm algorithm is

S = {(A, x , y) : x , y ∈ V and A ∈ Sx ,y}





x


Sx ,y = {A ⊆ E : ∂A = {x , y}}

Sx ,x = {A ⊆ E : ∂A = ∅}


◮ State space of worm algorithm is

S = {(A, x , y) : x , y ∈ V and A ∈ Sx ,y}

◮ Assign (A, x , y) ∈ S probability

π(A, x , y) ∝ dx dy (tanh β)|A|



Ising susceptibility◮ If Z :=

∑A∈C(G) w |A| and w := tanh β

ZIsing =(

2|V | cosh|E | β)

Z Partition function

Z 〈σxσy 〉Ising =∑

A∈Sx,y

w |A| Two-point function

Z 〈M2〉Ising =∑

A∈S

w |A| Magnetization





ZIsing =(

2|V | cosh|E | β)



A∈Sx,y



A∈S

w |A| Magnetization

◮ If G is translationally invariant then

π(A, x , y) = w |A|/Z V χ





ZIsing =(

2|V | cosh|E | β)



A∈Sx,y



A∈S

w |A| Magnetization

◮ If G is translationally invariant then

π(A, x , y) = w |A|/Z V χ

◮ Therefore the observable D0(A, x , y) = δx ,y satisfies

〈D0〉π = 1/χ



Worm dynamics

x

y◮ Start in configuration (A, x , y)



Worm dynamics

x


◮ Pick x or y



Worm dynamics

x x ′


◮ Pick x or y◮ Pick x ′ ∼ x



Worm dynamics

x

y

x ′◮ Start in configuration (A, x , y)

◮ Pick x or y◮ Pick x ′ ∼ x◮ Propose (A, x , y) → (A△xx ′, x ′, y)

◮ If transition removes an edge accept with probability 1



Worm dynamics

x

y

x ′◮ Start in configuration (A, x , y)


◮ If transition removes an edge accept with probability 1◮ If transition adds an edge accept with probability w



Worm dynamics

◮ Start in configuration (A, x , y)


◮ If transition removes an edge accept with probability 1◮ If transition adds an edge accept with probability w



Transition matrix

◮ Let G be translationally invariant with degree z◮ Worm dynamics corresponds to transition matrix P on S

P[(A, x , y) → (A△xx ′, x ′, y)] =12

1z

{1, xx ′ ∈ A,

w , xx ′ 6∈ A,

◮ And similarly for y moves. . .◮ All other non-diagonal elements of P are zero



Transition matrix

◮ Let G be translationally invariant with degree z◮ Worm dynamics corresponds to transition matrix P on S

P[(A, x , y) → (A△xx ′, x ′, y)] =12

1z

{1, xx ′ ∈ A,

w , xx ′ 6∈ A,

◮ And similarly for y moves. . .◮ All other non-diagonal elements of P are zero

LemmaP is in detailed balance with π(A, x , y) = w |A|/Z V χ

◮ Can estimate χ by running the worm dynamics



Efficiency

◮ Worm dynamics provide a valid way to compute χ



Efficiency


◮ But how efficient is the worm algorithm?



Efficiency


◮ But how efficient is the worm algorithm?◮ How do we measure efficiency anyway?



Efficiency


◮ But how efficient is the worm algorithm?◮ How do we measure efficiency anyway?◮ Empirically – measuring autocorrelations


Autocorrelations, critical slowing down . . .

Markov-chain Monte Carlo◮ Markov chain

◮ State space S, with |S| < ∞◮ Transition matrix P◮ Stationary distribution π





◮ Observables (random variables) X , Y , . . .






◮ Simulate Markov chain s0P−→ s1

P−→ s2

P−→ . . . with st ∈ S







P−→ s2

P−→ . . . with st ∈ S

◮ Get time series X0, X1, X2, . . . with Xt = X (st)







P−→ s2

P−→ . . . with st ∈ S


◮ Define the autocorrelation function

ρX (t) :=〈XsXs+t〉π − 〈X 〉2

π

varπ(X )







P−→ s2

P−→ . . . with st ∈ S


◮ Define the autocorrelation function

ρX (t) :=〈XsXs+t〉π − 〈X 〉2

π

varπ(X )

◮ Stationary process – start “in equilibrium”



Integrated autocorrelation times

◮ The integrated autocorrelation time

τint,X :=12

∞∑

t=−∞

ρX (t)





τint,X :=12

∞∑

t=−∞

ρX (t)

◮ If X̂ is the sample mean of {Xt}Tt=1 then we have

var(X̂ ) ∼ 2 τint,Xvar(X )

T, T → ∞





τint,X :=12

∞∑

t=−∞

ρX (t)

◮ If X̂ is the sample mean of {Xt}Tt=1 then we have

var(X̂ ) ∼ 2 τint,Xvar(X )

T, T → ∞

◮ 1 “effectively independent” observation every 2 τint,X steps



Exponential autocorrelation times

◮ ρX (t) typically decays exponentially as t → ∞

◮ The exponential autocorrelation time

τexp,X := lim supt→∞

t− log |ρX (t)|

and τexp := supX

τexp,X







t− log |ρX (t)|

and τexp := supX

τexp,X

◮ Typically τexp,X = τexp < ∞ and τint,X ≤ τexp for all X







t− log |ρX (t)|

and τexp := supX

τexp,X


◮ Start the chain with arbitrary distribution α

◮ Distribution at time t is αP t







t− log |ρX (t)|

and τexp := supX

τexp,X


◮ Start the chain with arbitrary distribution α

◮ Distribution at time t is αP t

LemmaαP t tends to π with rate bounded by e−t/τexp



Critical slowing-down

◮ Near a critical point the autocorrelation times typicallydiverge like

τ ∼ ξz





τ ∼ ξz

◮ More precisely, we have a family of exponents:zexp, and zint,X for each observable X .





τ ∼ ξz

◮ More precisely, we have a family of exponents:zexp, and zint,X for each observable X .

◮ Different algorithms for the same model can have verydifferent z

◮ E.g. d = 2 Ising model◮ Glauber (Metropolis) algorithm z ≈ 2◮ Swendsen-Wang algorithm z ≈ 0.2


Numerical results

Worm simulations

◮ Simulated the critical square-lattice Ising model


Numerical results

Worm simulations

◮ Simulated the critical square-lattice Ising model◮ Focus on two observables:

◮ N (A, x , y) = |A|◮ D0(A, x , y) = δx,y

◮ 〈N〉 is “energy-like”◮ 〈D0〉 = 1/χ


Numerical results

Worm simulations

◮ Simulated the critical square-lattice Ising model◮ Focus on two observables:

◮ N (A, x , y) = |A|◮ D0(A, x , y) = δx,y

◮ 〈N〉 is “energy-like”◮ 〈D0〉 = 1/χ

◮ Measured observables after every hit (worm update)◮ Natural unit of time is one sweep (Ld hits)


Numerical results

Dynamics of N

◮ ρN (t) is almost a perfect exponential

0.01

0.1

1

0 1 2 3 4 5 6

ρ N (

t)

t/τint,N

0.01

0.1

1

0 1 2 3 4 5 6

ρ N (

t)

t/τint,N

8 16 32 64

128 256 512

10242048

◮ Scaled time by τint,N

◮ Good data collapse suggestszexp ≈ zint,N

◮ Fitting τint,N gives zint,N ≈ 0.379

◮ Li-Sokal bound zexp, zint,N ≥ α/ν applies to worm dynamics


Numerical results

Dynamics of D0

◮ ρD0(t) decays significantly in O(1) hits!

10-6

10-5

10-4

10-3

10-2

10-1

1

10710610510410310210 1

ρ D0 (

t)

t

L increases

10-6

10-5

10-4

10-3

10-2

10-1

1

10710610510410310210 1

ρ D0 (

t)

t

L increases

8 16 32 64

128 256 512

10242048 ◮ ρD0(t) ∼ t−s with s ≈ 0.75

◮ D0 decorrelates on a totally different time scale to N


Numerical results

Three dimensions

◮ Qualitatively similar behavior when d = 3:


Numerical results

Three dimensions

◮ Qualitatively similar behavior when d = 3:◮ ρD0(t) ∼ t−s

◮ s ≈ 0.66


Numerical results

Three dimensions


◮ s ≈ 0.66◮ ρN (t) roughly exponential◮ zexp ≈ zint,N ≈ α/ν ≈ 0.174◮ Li-Sokal bound may be sharp for d = 3 worm algorithm


Numerical results

Three dimensions


◮ s ≈ 0.66◮ ρN (t) roughly exponential◮ zexp ≈ zint,N ≈ α/ν ≈ 0.174◮ Li-Sokal bound may be sharp for d = 3 worm algorithm◮ Compare Swendsen-Wang zSW ≈ 0.46


Numerical results

Practical efficiency for square/cubic lattice critical Ising

◮ Swendsen-Wang seems to outperform worm when d = 2


Numerical results


◮ Swendsen-Wang seems to outperform worm when d = 2◮ Efficiency depends on observable, X


Numerical results


◮ Swendsen-Wang seems to outperform worm when d = 2◮ Efficiency depends on observable, X◮ A simple way to compare worm and SW is to compute

κ = TCPU varX̂ for both algorithms


Numerical results



κ = TCPU varX̂ for both algorithms◮ When d = 3 and X = D0 we find κworm/κSW ≈ L−0.33

◮ With the crossover κworm/κSW ≈ 1 at around L ≈ 20


Numerical results





◮ There is also a natural worm estimator for ξ


Numerical results






◮ Again SW outperforms worm when d = 2


Numerical results






◮ Again SW outperforms worm when d = 2◮ For d = 3 we find κworm/κSW ≈ L−0.32



An alternate perspective on worm dynamics

An alternative perspective on worm dynamics◮ Our perspective so far:

Worm algorithm ⇐⇒ simulate Ising high-temperature graphs





◮ Eulerian-subgraph model





◮ Eulerian-subgraph model◮ State space C(G)





◮ Eulerian-subgraph model◮ State space C(G)◮ P(A) ∝ w |A|






◮ Run a worm simulation






◮ Run a worm simulation◮ Only observe the chain when A ∈ C(G), i.e. when x = y






◮ Run a worm simulation◮ Only observe the chain when A ∈ C(G), i.e. when x = y◮ Obtain a new Markov chain P◮ Stationary distribution π(A, x , x) ∝ f (x) w |A|






◮ Run a worm simulation◮ Only observe the chain when A ∈ C(G), i.e. when x = y◮ Obtain a new Markov chain P◮ Stationary distribution π(A, x , x) ∝ f (x) w |A|

◮ New perspective:

Worm algorithm ⇐⇒ simulate Eulerian-subgraph model



Loop models

◮ Honeycomb lattice Eulerian subgraphs = disjoint cycles◮ Nienhuis loop model



Loop models


0 < w < wc Disordered phasewc < w < ∞ Critical densely-packed phase

w = +∞ Critical fully-packed phase



Loop models




+ + + ++ − − + +

+ − − ++ − + − +

+ − − ++ + + + + ◮ 1 loop config ↔ 2 dual spin configs

◮ Ising spin domains



Loop models




+ + + ++ − − + +

+ − − ++ − + − +


◮ Ising spin domains◮ w = e−2β



Loop models




+ + + ++ − − + +

+ − − ++ − + − +



◮ w > 1 =⇒ antiferromagnetic β



Loop models




+ + + ++ − − + +

+ − − ++ − + − +



◮ w > 1 =⇒ antiferromagnetic β

◮ Worm ⇐⇒ simulate dual Ising domain boundaries



Fully-packed loop model (FPL)




◮ FPL ↔ triangular-lattice Ising antiferromagnet at T = 0




+−

+

Frustration

◮ FPL ↔ triangular-lattice Ising antiferromagnet at T = 0◮ Frustrated systems hard to simulate




+−

+

Frustration

◮ FPL ↔ triangular-lattice Ising antiferromagnet at T = 0◮ Frustrated systems hard to simulate◮ Cluster algorithms for frustrated Ising models thought to be

non-ergodic at T = 0




+−

+

Frustration

◮ FPL ↔ triangular-lattice Ising antiferromagnet at T = 0◮ Frustrated systems hard to simulate◮ Cluster algorithms for frustrated Ising models thought to be

non-ergodic at T = 0◮ Can we use worm instead?



P∞ has absorbing states

◮ Standard worm transitions for w ≥ 1 on z-regular graph:

Pw [(A, x , y) → (A△xx ′, x ′, y)] =1

2z

{1/w xx ′ ∈ A

1 xx ′ 6∈ A





Pw [(A, x , y) → (A△xx ′, x ′, y)] =1

2z

{1/w xx ′ ∈ A

1 xx ′ 6∈ A

◮ Identity transitions are fixed by normalization:

Pw [(A, x , y) → (A, x , y)] = (1 − 1/w)dx(A) + dy (A)

2z





Pw [(A, x , y) → (A△xx ′, x ′, y)] =1

2z

{1/w xx ′ ∈ A

1 xx ′ 6∈ A



2z

◮ P∞[(A, x , y) → (A, x , y)] = 1 whenever dx(A) = dy (A) = z





Pw [(A, x , y) → (A△xx ′, x ′, y)] =1

2z

{1/w xx ′ ∈ A

1 xx ′ 6∈ A



2z

◮ P∞[(A, x , y) → (A, x , y)] = 1 whenever dx(A) = dy (A) = z◮ Cannot use standard worm algorithm when w = +∞



How do we solve the problem?

◮ P∞[(A, x , y) → (A, x , y)] = 1 whenever dx(A) = dy (A) = z




◮ P∞[(A, x , y) → (A, x , y)] = 1 whenever dx(A) = dy (A) = z◮ But on the honeycomb lattice z = 3




◮ P∞[(A, x , y) → (A, x , y)] = 1 whenever dx(A) = dy (A) = z◮ But on the honeycomb lattice z = 3◮ So states with Eulerian A never get stuck




◮ P∞[(A, x , y) → (A, x , y)] = 1 whenever dx(A) = dy (A) = z◮ But on the honeycomb lattice z = 3◮ So states with Eulerian A never get stuck◮ To simulate loop models, we only observe the chain when

A is Eulerian. . .





A is Eulerian. . .◮ Try defining new transition matrix P ′

∞ such that:






∞ such that:◮ P ′

∞[(A, x , x) → · ] = limw→∞ Pw [(A, x , x) → · ]







∞[(A, x , x) → · ] = limw→∞ Pw [(A, x , x) → · ]◮ P ′

∞[(A, x , y) → (A, x , y)] = 0 when x 6= y







∞[(A, x , x) → · ] = limw→∞ Pw [(A, x , x) → · ]◮ P ′

∞[(A, x , y) → (A, x , y)] = 0 when x 6= y

◮ This will get rid of the absorbing states. . .







∞[(A, x , x) → · ] = limw→∞ Pw [(A, x , x) → · ]◮ P ′

∞[(A, x , y) → (A, x , y)] = 0 when x 6= y

◮ This will get rid of the absorbing states. . .◮ If we are lucky P ′

∞ will simulate the FPL. . .



Worm algorithm for honeycomb-lattice FPL◮ Start in configuration (A, x , y)




◮ If x = y




◮ If x = y◮ Pick x ′ ∼ x




◮ If x = y◮ Pick x ′ ∼ x◮ If xx ′ 6∈ A

◮ (A, x , x) → (A ∪ xx ′, x ′

, x) or(A, x , x) → (A ∪ xx ′

, x , x ′)





◮ (A, x , x) → (A ∪ xx ′, x ′

, x) or(A, x , x) → (A ∪ xx ′

, x , x ′)

◮ Else (A, x , x) → (A, x , x)





◮ (A, x , x) → (A ∪ xx ′, x ′

, x) or(A, x , x) → (A ∪ xx ′

, x , x ′)

◮ Else (A, x , x) → (A, x , x)

◮ Else if x 6= y





◮ (A, x , x) → (A ∪ xx ′, x ′

, x) or(A, x , x) → (A ∪ xx ′

, x , x ′)

◮ Else (A, x , x) → (A, x , x)

◮ Else if x 6= y◮ Pick x or y (say x)





◮ (A, x , x) → (A ∪ xx ′, x ′

, x) or(A, x , x) → (A ∪ xx ′

, x , x ′)

◮ Else (A, x , x) → (A, x , x)

◮ Else if x 6= y◮ Pick x or y (say x)◮ If dx(A) = 3

◮ Pick one of the three xx ′ ∈ A◮ (A, x , y) → (A \ xx ′

, x ′, y)





◮ (A, x , x) → (A ∪ xx ′, x ′

, x) or(A, x , x) → (A ∪ xx ′

, x , x ′)

◮ Else (A, x , x) → (A, x , x)

◮ Else if x 6= y◮ Pick x or y (say x)◮ If dx(A) = 3

◮ Pick one of the three xx ′ ∈ A◮ (A, x , y) → (A \ xx ′

, x ′, y)

◮ Else if dx(A) = 1◮ Pick one of the two xx ′ 6∈ A◮ (A, x , y) → (A ∪ xx ′

, x ′, y)



Transition matrix◮ x 6= y

P ′∞[(A, x , y) → (A△xx ′, x ′, y)] =

{1/6 dx(A) = 3

1/4 xx ′ 6∈ A◮ x = y

P ′∞[(A, x , x) → (A, x , x)] =

dx(A)

3

P ′∞[(A, x , x) → (A ∪ xx ′, x ′, x)] =

16




P ′∞[(A, x , y) → (A△xx ′, x ′, y)] =

{1/6 dx(A) = 3

1/4 xx ′ 6∈ A◮ x = y

P ′∞[(A, x , x) → (A, x , x)] =

dx(A)

3

P ′∞[(A, x , x) → (A ∪ xx ′, x ′, x)] =

16

TheoremThe set of states in S with no isolated vertices is recurrent andirreducible, its complement is transient, and the stationarydistribution of P ′

∞ is uniform on the fully-packed configurations




P ′∞[(A, x , y) → (A△xx ′, x ′, y)] =

{1/6 dx(A) = 3

1/4 xx ′ 6∈ A◮ x = y

P ′∞[(A, x , x) → (A, x , x)] =

dx(A)

3

P ′∞[(A, x , x) → (A ∪ xx ′, x ′, x)] =

16

TheoremThe set of states in S with no isolated vertices is recurrent andirreducible, its complement is transient, and the stationarydistribution of P ′

∞ is uniform on the fully-packed configurations◮ Therefore P ′

∞ correctly simulates the FPL


Numerical results

Worm simulations of honeycomb-lattice FPL

◮ Simulated the honeycomb-lattice FPL


Numerical results


◮ Simulated the honeycomb-lattice FPL◮ Again observed multiple time scales


Numerical results


◮ Simulated the honeycomb-lattice FPL◮ Again observed multiple time scales◮ Nl(A, x , y) = number of loops (cyclomatic number)◮ 〈Nl〉 is “energy-like”


Numerical results


◮ Simulated the honeycomb-lattice FPL◮ Again observed multiple time scales◮ Nl(A, x , y) = number of loops (cyclomatic number)◮ 〈Nl〉 is “energy-like”◮ Nl is the slowest mode observed


Numerical results

Dynamics of Nl

◮ ρNl (t) is almost a perfect exponential

0.01

0.1

1

0 1 2 3 4 5 6 7 8

ρ Nl (

t)

t/τint,Nl

0.01

0.1

1

0 1 2 3 4 5 6 7 8

ρ Nl (

t)

t/τint,Nl

12 24 30 60 90150240600900

◮ Scaled time by τint,Nl

◮ Good data collapse suggestszexp ≈ zint,Nl

◮ Fitting τint,Nl giveszint,Nl = 0.515(8)


Summary

◮ Standard worm algorithm for Ising high-temperaturegraphs outperforms SW in three dimensions


Summary


◮ Worm decorrelates on different time scales – depends onobservable


Summary



◮ Modified worm algorithm is demonstrated to be valid forhoneycomb lattice FPL

◮ Dynamic exponent z ≈ 0.5


Summary



◮ Modified worm algorithm is demonstrated to be valid forhoneycomb lattice FPL

◮ Dynamic exponent z ≈ 0.5◮ By contrast, even the best cluster algorithms for frustrated

Ising models are thought to be non-ergodic at T = 0

worm algorithms - ustc

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