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Replica Monte Carlo Simulation Jian-Sheng Wang National University of Singapore: 

Replica Monte Carlo Simulation Jian-Sheng Wang National University of Singapore


Outline Review of extended ensemble methods (multi-canonical, Wang-Landau, flat-histogram, simulated tempering) Replica MC Connection to parallel tempering and cluster algorithm of Houdayer Early and new results

Slowing Down at First-Order Phase Transition: 

Slowing Down at First-Order Phase Transition At first-order phase transition, the longest time scale is controlled by the interface barrier where β=1/(kBT), σ is interface free energy, d is dimension, L is linear size

Multi-Canonical Ensemble: 

Multi-Canonical Ensemble We define multi-canonical ensemble as such that the (exact) energy histogram is a constant h(E) = n(E) f(E) = const This implies that the probability of configuration is P(X)  f(E(X))  1/n(E(X))

Multi-Canonical Simulation (Berg et al): 

Multi-Canonical Simulation (Berg et al) Do simulation with probability weight fn(E), using Metropolis acceptance rate min[1, fn(E’)/fn(E) ] Collection histogram H(E) Re-compute weight by fn+1(E) = fn(E)/H(E) Iterate until H(E) is flat

Multi-Canonical Simulation and Reweighting: 

Multi-Canonical Simulation and Reweighting Multicanonical histogram and reweighted canonical distribution for 2D 10-state Potts model From A B Berg and T Neuhaus, Phys Rev Lett 68 (1992) 9.

Wang-Landau Method: 

Wang-Landau Method Work directly with n(E), starting with some initial guess, n(E) ≈ const, f = f0 > 1 (say 2.7) Flip a spin according to acceptance rate min[1, n(E)/n(E ’)] And also update n(E) by n(E) <- n(E) f Reduce f by f <-f 1/2 after certain number of MC steps, when the histogram H(E) is “flat”.

Flat Histogram Algorithm: 

Flat Histogram Algorithm Pick a site at random Flip the spin with probability where E is current and E ’ is new energy 3. Accumulate statistics for <N(σ,E ’-E)>E

The Ising Model: 

The Ising Model - + + + + + + + + + + + + + + + + - - - - - - - - - - - - - - - - - - - Total energy is E(σ) = - J ∑<ij> σi σj sum over nearest neighbors, σi = ±1 N(s,DE) is the number of sites, such that flip spin costs energy DE. σ = {σ1, σ2, …, σi, … } DE=0 DE=-8J

Spin Glass Model: 

Spin Glass Model + + + + + + + + + + + + + + + + + + + + + + + + - - - - - - - - - - - - - - - - - - - - - - - - - A random interaction Ising model - two types of random, but fixed coupling constants (ferro Jij > 0) and (anti-ferro Jij < 0)

Slow Dynamics in Spin Glass: 

Slow Dynamics in Spin Glass Correlation time in single spin flip dynamics for 3D spin glass. t  |T-Tc|6. From Ogielski, Phys Rev B 32 (1985) 7384.

Tunneling Time for 3D Spin Glass: 

Tunneling Time for 3D Spin Glass Diamond: standard flat histogram algorithm; dot: with N-fold way; triangle: equal-hit algorithm. From J S Wang & R H Swendsen, J Stat Phys, 106 (2002) 245.

First-Passage Time to Ground States: 

First-Passage Time to Ground States Number of sweeps needed to discover a ground state for the first time. Extremal Optimization (EO) is an optimization algorithm. From J S Wang and Y Okabe, J Phys Soc Jpn, 72 (2003) 1380.

Simulated Tempering (Marinari & Parisi, 1992): 

Simulated Tempering (Marinari & Parisi, 1992) Simulated tempering treats parameters as dynamical variables, e.g., β jumps among a set of values βi. We enlarge sample space as {X, βi}, and make move {X,βi} -> {X’,β’i} according to the usual Metropolis rate.

Replica Monte Carlo: 

Replica Monte Carlo A collection of M systems at different temperatures is simulated in parallel, allowing exchange of information among the systems. β1 β2 β3 βM . . .

Moves between Replicas: 

Moves between Replicas Consider two neighboring systems, σ1 and σ2, the joint distribution is P(σ1,σ2)  exp[-β1E(σ1) –β2E(σ2)] = exp[-Hpair(σ1, σ2)] Any valid Monte Carlo move should preserve this distribution

Pair Hamiltonian in Replica Monte Carlo: 

Pair Hamiltonian in Replica Monte Carlo We define i=σi1σi2, then Hpair can be rewritten as The Hpair again is a spin glass. If β1≈β2, and two systems have consistent signs, the interaction is twice as strong; if they have opposite sign, the interaction is 0.

Cluster Flip in Replica Monte Carlo: 

Cluster Flip in Replica Monte Carlo  = +1  = -1 Clusters are defined by the values of i of same sign, The effective Hamiltonian for clusters is Hcl = - Σ kbc sbsc Where kbc is the interaction strength between cluster b and c, kbc= sum over boundary of cluster b and c of Kij. b c Metropolis algorithm is used to flip the clusters, i.e., σi1 -> -σi1, σi2 -> -σi2 fixing  for all i in a given cluster.

Apply Swendsen-Wang in Replica MC: 

Apply Swendsen-Wang in Replica MC The t-cluster can be further broken down. Within a t-cluster, a bond is set with probability P = 1 – exp(-2 (b1+b2)|Jij|) if interaction is satisfied Jijsisj > 0; no bond otherwise. No interaction between clusters broken this way.  = +1  = -1 b c

Implementation Issues: 

Implementation Issues Use Hoshen-Kompelman algorithm to identify clusters Based on cluster size and total number of clusters, pre-allocate memory to store effective cluster coupling kab Order O(N) algorithm for each sweep

Comparing Correlation Times: 

Comparing Correlation Times Correlation times as a function of inverse temperature K=βJ on 2D, ±J Ising spin glass of 32x32 lattice. From R H Swendsen and J-S Wang, Phys Rev Lett 57 (1986) 2607. Replica MC Single spin flip

Cluster Algorithm of S Liang: 

Cluster Algorithm of S Liang 2D Gaussian spin glass on 16x16 lattice, using a generalization due to F Niedermayer. From S Liang, PRL 69 (1992) 2145.

Replica Exchange (Hukushima & Nemoto, 1996): 

Replica Exchange (Hukushima & Nemoto, 1996) A simple move of exchange configurations, σ1 <-> σ2, with Metropolis acceptance rate min{ 1, exp[(β2-β1)(E(σ2)-E(σ1))] } This is equivalent to flip all the ti =-1 clusters in replica Monte Carlo. Also known as parallel tempering

Replica Exchange: 

Replica Exchange Spin-spin exponential relaxation time for replica exchange on 123 lattice. From K Hukushima and K Nemoto, J Phys Soc Jpn, 65 (1996) 1604.

Houdayer’s Cluster Algorithm: 

Houdayer’s Cluster Algorithm β1 β2 β3 βM . . . β1 β2 β3 βM . . . β1 β2 β3 βM . . . . . . Replica exchange between different temperatures Single t-cluster flip between same temperature set 1 set 2 set N Simulate simultaneously M by N systems.

Relaxation towards Equilibrium at LowT: 

Relaxation towards Equilibrium at LowT Relaxation of energy for 100x100 +/-J Ising spin glass at T = 0.1 [32 set of 26 replicas for cluster algorithm]. From J Houdayer, Eur Phys J B 22 (2001) 479.

Correlation Functions in Replica MC: 

Correlation Functions in Replica MC Time correlation function for order parameter q on 128x128 ±J Ising spin glass. 106 MCS used. Labels are K=1/T. q=|∑ii| From J-S Wang and R H Swendsen, cond-mat/0407273.

Comparison of Single-spin-flip, Parallel Tempering, Houdayer, and Replica MC: 

Comparison of Single-spin-flip, Parallel Tempering, Houdayer, and Replica MC 2D ±J Ising spin glass integrated correlation time on a 32x32 lattice. From cond-mat/0407273, to appear (2005) Prog Theor Phys Suppl.

Integrated Correlation Times, 128x128 system: 

Integrated Correlation Times, 128x128 system

Comparison in 3D: 

Comparison in 3D Integrated correlation times for ±J Ising spin glass on 12x12x12 lattice.

2D Spin Glass Susceptibility: 

2D Spin Glass Susceptibility 2D ±J spin glass susceptibility on 128x128 lattice, 1.8x104 MC steps. From J S Wang and R H Swendsen, PRB 38 (1988) 4840.   K5.11 was concluded.

Heat Capacity at Low T: 

Heat Capacity at Low T c  T -2exp(-2J/T) This result is confirmed recently by Lukic et al, PRL 92 (2004) 117202. slope = -2

Monte Carlo Renormalization Group : 

Monte Carlo Renormalization Group YH defined by with RG iterations for difference sizes in 2D. From J S Wang and R H Swendsen, PRB 37 (1988) 7745.

MCRG in 3D: 

MCRG in 3D 3D result of YH. MCS is 104 to 105, with 23 samples for L= 8, 8 samples for L= 12, and 5 samples for L= 16.

Correlation Length: 

Correlation Length Correlation length (defined by ratio of wavenumber dependent susceptibilities) on 128x128 lattice, averaged of 96 random coupling samples. Unpublished.


Summary Replica MC is very efficient in 2D, and becomes equivalent to Parallel Tempering in 3D Replica MC has been used for equilibrium simulations (heat capacity, MCRG, etc)

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