09/14/2008

Markov Chain Monte Carlo: innovations and applications in

statistical physicsYoujin Deng

With Alan D. Sokal (NYU)Henk W.J. Bloete (Leiden)Timothy G Garoni (NYU)Wenan Guo (BNU)

Hefei

09/14/2008

Content

• Models: Potts and RC

• Introduction: Markov Chain Monte Carlo (MCMC)

• Local algorithms: Metropolis, Sweeny, and Worm

• Collective–mode algorithms: Swendsen-Wang (SW)

• Three pictures for SW method• Applications: real q>1 RC model, RC model in field,

Loop, Anti-Potts, AT, fixed-bond RC, fixed-magnetization Potts, spin-glass

• Some references (incomplete and biased)

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Model

• Potts modelHamiltonian:

Partition sum:

• Random-cluster model

( , ) ( 1,2, )i jij

H K qδ σ σ σ⟨ ⟩

= − =∑ L

( ) ( )b G k G

G

Z v q= ∑

/H kTZ e−= ∑

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Markov Chain Monte Carlo(MCMC)

• desired probability distribution ;transition probability matrix

• detailed balancing:• irreducibility (ergodicity)

1( , )t tT +Γ Γ

( )P Γ

( , ') ( ') ( ', ) ( )T P T PΓ Γ Γ = Γ Γ Γ

Efficiency: Critical Slowing-down

zτ ξ∝

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Local algorithms

• Metropolis for Ising

1), pick up a spin (randomly or sequentially)2), calculate the energy cost if the spin is flipped3), flip the spin with probabilityDynamic exp:

• Sweeny (two-time scaling): critical speeding-up!

• Worm (three-time scaling)

EΔ/Min(1, )E kTe−Δ

2.2z ≈

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Swendsen-Wang (SW) Method

Swendsen-Wang Simulation:1), for each edge , if , place a bond with

; otherwise, do nothing.

2), for each connected component (FK cluster), randomly pick up one of the q states.

ijei jσ σ=

ije

1 Kp e−= −

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Three Pictures for SW Method

• Edward-Sokal Picture1), Exact mapping between the Potts model and the random-cluster (RC) model:

2), Bond-Spin-joint probability measure:

! SW method passes back and forward between the bond and thespin representation of the Potts model.

( ) | | | | | |(1 )k A E A A

A EZ q p p−

⊆

= −∑

,0 ,1( , ) [(1 ) ( )]e en n e

e E

P n p pσ δ δ δ σ∈

∝ − +∏r

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• Domany’s Picture1), Hamiltonian: ; is a

two-energy-level system.2), Probability measure

with

3), for unit i in energy , place bond with .

4), Perform operations that conserve energies of units i with bonds. (do-nothing is surely one valid operation)

1 2( ) or iH E Eσ =r

ii

H H= ∑

2

1,exp( ) (1 )i

Ei i H EH e v δ−− = +

2 1( 1)E Eiv e −= −

1E /(1 )i i ip v v= +

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• Induced subgraph Picture1, RC model

2, Coloring: Independently for each component, assignit a “color” with probability . Vertex set ispartitioned as . Conditioning on the color assignment, independently on each induced subgraph

is a -state RC model.3, Choose any MC method to update the induced RC model. Particularly, it is a bond percolation for q=1. Do nothing is also a valid update.

( ) ( )

11 1

k A k A m

e e mA E A Ee E i e E i

Z v q v qα⊆ ⊆ =∈ = ∈ =

= =∑ ∑ ∑∏ ∏ ∏ ∏

α /q qα V1

m

V Vαα =

= U

[ ]G Vα qα

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Application• Chayes-Machta Method for q>1 RC Model

1, RC Model

2, Independently for each component, color it to be “1”with , and color it be “2” with .

3, Update subgraph as the bond percolation, and do nothing for .

| | ( ) | | ( )1 2 1 2[ ] ( 1, 1)A k A A k A

A E A EZ v q v q q q q q

⊆ ⊆

= = + = = −∑ ∑

1/p q= ( 1) /p q q= −

1[ ]G V2[ ]G V

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• Potts Model in a field

1, Partition sum

2, Independently for each component, color it to be “1”with ; otherwise, color it be “2”.

3, Update subgraph as the bond percolation, and do nothing for .

( )| |

,11

exp( ) [( 1) | | ]e j

k AA h

iA Ee E j V i

Z K h v q Vσ σσ

δ δ⊆∈ ∈ =

= = − +∑ ∑∏ ∏ ∏

1 | | /[( 1) | | ]h hi ip V q V= − +

1[ ]G V

2[ ]G V

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• Loop Model (Honeycomb)1, Partition sum

c(A): Loop (cyclomatic) number2, Physical relevance:

(a), it is the high-T graph of Nienhuis’s O(n) spin model.For q=1, it is a graph representation of Ising model.

(b), for , it reduces to the self-avoiding randomwalk (SAW).

(c), it plays an important role in the stochastic Loewnerevolution(SLE)

| | | ( )|

:Eulerian

A c A

A E

Z x n⊆

= ∑

0q →

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• Simulation for n=1 (Honeycomb)

1, Plaque update:

2, Worm algorithm

3, Cluster simulation of the Ising-spin model on triangular lattice with coupling . (Duality relation between high-T and Low-T graphs)

*2Ke x− =

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• Cluster Simulation for n>1 (Honeycomb)Start from bond config. on and spin config. on . 1, Color each loop (cycle) to be “1” with and to be “2” with . Color isolated sites to be “1”.

2, Place bonds on each edge on triangular lattice . If the dual does not entirely lie in , place a bond; otherwise, place a bond with .

3, Form clusters on . Independently for each component, flip the Ising spins with .

4, New bond config. on is the low-T graph of spins on

! Analogous idea applies to face-/corner-cubic model

T

1/p n=1 1/p n= −

*ee

1V1p x= −

T1/ 2p =

H T

H T

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• Antiferromagnetic Potts model (Domany picture):

1, Edge weight:

2, Choose two of the q states—say . Place bond with on edges connecting states .

3, Form clusters. Independently for each cluster, interchange Potts states with

,exp( ) [1 ( 1)(1 )]i j i j

K KK e eσ σ σ σδ δ−= + − −

1 Kp e= −

1/ 2.p =

1 2and q q1 2and q q

1 2q q↔

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• Antiferromagnetic triangular Ising model:1, Hamiltonian:

(two satisfied and one unsatisfied bond).

(three unsatisfied bonds).

2, weight:

3, For each , place a bond with on one of the two satisfied bonds. 4, For clusters, and update Ising spins.

41 Kp e= −

i

ij i

i je E

H Ks s HΔ∈ Δ

= − =−∑ ∑

H KΔ =3H KΔ = −

Δ 3 4,exp( ) [1 ( 1) ]K K

H KH e e δΔ

−Δ− = + −

Δ

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• Fixed-bond-number RC model1, Definition:

2, For each cluster, color it to be “1” with , and color it be “2” with .

3, On subgraph , do Kawasaki dynamic for bond percolation, and do nothing for .

| | ( )

;| | const.

A k A

A E AZ v q

⊆ =

= ∑1/p q=

1 1/p q= −

1[ ]G V

2[ ]G V

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• Geometric Cluster algorithm (Ising Model)

Let vertices map onto under certain transformation—i.e., the spatial inversion. Under interchanging-spin operation , the energy associated with edges has two levels:

Say , weight:

2 2 1

' ' 1, ,exp[ ] [1 ( 1) ]ij i j e

E E Ee e H EH e e δ− −− = + −

, ,i j k ', ', 'i j k

'i is s↔' 'and ij i je e

' ' 1

' ' 2

( ) : or

( ) :e i j i j

e i j i j

H K s s s s E

H K s s s s E

= − + ≡

= − + ≡

1 2E E<

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• Geometric Cluster algorithm (Ising Model)(Single-Cluster version)1, randomly chose a site . Let and its mapping

be in the cluster, and do operation .

2, for all neighbor sites of (not yet in cluster):If , place a bond with , doand include in the cluster (stack). Otherwise, do nothing.

3, read a site from stack, do Step 2. Erase from stack.

4, Repeat Steps 2 and 3 until stack is empty.

i i 'i'i is s↔

k i

1, 1eH Eδ = 1 21 E Ep e −= − 'k ks s↔

, 'k k

j j

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• Embedding methods for a variety of systems.1), O(n) spin model2), Ashkin-Teller model3), Baxter-Wu model….

• Replica and simulated tempering MC for spin glass

• SW-like MC algorithm for quantum Potts model

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Some references• M.E.J. Newman and G.T. Barkema, Monte Carlo Methods in Statistical

Physics (Clarendon Press, 1999).• K. Binder and D.W. Heermann, Monte Carlo Simulation in Statistical

Physics: An Introduction, 4th ed. (Springer-Verlag, 2002).• N. Metropolis et.al, JCP 21, 1087 (1953).• R.H. Swendsen and J.-S. Wang, PRL 58, 86 (1987); 57, 2067 (1986)• J.-S. Wang, R.H. Swendsen and R. Koteck´y, PRB 42, 2465 (1990).• R.G. Edwards and A.D. Sokal, PRD 38, 2009 (1988).• D. Kandel, R. Ben-Av, and E. Domany, PRB 45 4700 (1992).• Y.J. Deng et.al, PRL 88, 190602 (2002), 98, 030602 (2007), 98 120601

(2007), 98 230602 (2007), 99, 110601 (2007), 99, 055701 (2007)• E. Marinari and G. Parisi, EPL 19, 451 (1992)• N. Prokof’ev and B. Svistunov, PRL 87, 160601 (2001).• J. R. Heringa and H. W. J. Blöte, PRE 57, 4976 (1998).• J.W. Liu and E. Luijten, PRL 92, 035504 (2004); 93, 0247802 (2004);

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