Locally converging algorithms for determining the critical temperature in Ising systems

Abstract

We introduce a class of algorithms that converge to criticality automatically, in a way similar to the invaded cluster algorithm. Unlike the invaded cluster algorithm which uses global percolation as a test for criticality, these local algorithms use an average over local observables, specifically the number of satisfied bonds, in a feedback loop which drives the system toward criticality. Two specific algorithms are introduced, the average algorithm and the locally converging Wolff algorithm. We apply these algorithms to study the Ising square lattice and the Ising Bethe lattice. We find reasonable convergence to the critical temperature for both systems under the locally converging Wolff algorithm. We also re-examine the phase diagram of the dilute twodimensional ͑2D͒ Ising model and find results supporting our previously reported conclusions regarding the existence of a local regime of magnetization below the percolations threshold. In addition, the presented algorithms are computationally more efficient than the invaded cluster algorithm, requiring less CPU time and memory.A useful technique in the computational study of phase transitions in spin systems is the invaded cluster algorithm ͑ICA͒. 1-5 This algorithm, and others of the same general approach, 6,7 have the property that without prior knowledge of the critical temperature they evolve a spin system to the vicinity of the critical temperature. Let us briefly review the ICA algorithm. Starting in an arbitrary spin state, for example the positive aligned state, the ICA forms one by one a collection of satisfied bonds from the set of all satisfied bonds associated with the spin state. After each bond is added to a cluster, the ICA tests whether this cluster spans the lattice, i.e., has an extent that is of the order of the size of the system. When such a spanning cluster is found, the ICA performs the final step of the Swendsen-Wang ͑SW͒ algorithm 8 using this set of clusters. This consists of assigning a random spin value to each of the clusters formed-for the two-state Ising model considered in this paper, this is equivalent to flipping each cluster's spin with probability 1/2.The SW algorithm relies on the relationship p͑T c ͒ =1 − e −2J/k b T c , established by Fortuin and Kasteleyn, 9 between the critical temperature T c of the Potts spin lattice with exchange coupling J and the critical bond probability p of the associated ͑multispecies͒ bond percolation problem. When the ICA algorithm acts on a spin state typical of T Ͻ T c , because the state is relatively ordered, the fraction of satisfied bonds needed to achieve percolation is lower than the fraction needed at a temperature of T c . Through the FortuinKastelyn relationship, this small fraction of satisfied bonds results in the execution of a SW Monte Carlo step at a high temperature. In a similar way, the ICA generates a large fraction of satisfied bonds when applied to a spin state at T Ͼ T c , and so executes a SW step at a temperature T Ͻ T c . As the ICA algorithm is repeated, the temperatur...

Critical point determination from probability distribution functions in the three dimensional Ising model

“…• Algorithms that not required to fine tune any parameter, examples of those kind are the Invasion Cluster Algorithm [2], algorithms based on Self Organisation [3] or the Locally Cluster Algorithm [4].…”