Dynamic canonical and microcanonical transition matrix analyses of critical behavior

Abstract

By monitoring the sampling of states with different magnetizations in transition matrix procedures a family of accurate and easily implemented techniques are constructed that automatically control the variation of the temperature or energy as the calculation proceeds. The accuracy of the method for a single Markov chain exceeds that of standard transition matrix procedures that accumulate elements from multiple chains.

Transition matrix cluster algorithms

“…Various techniques have accordingly been advanced for optimizing transition matrix calculations. These include accumulating transitions from multiple independent Markov chains and dynamically adapting the temperature schedule to variations in the size of the accessible state space [15] by monitoring either the correlation time, [28] the convergence of the normalized histogram of samples as a function of magnetization [35] or the canonical entropy of either the full phase space or the phase space in the magnetization-energy diagram. [36] Additionally, the procedure can be accelerated through a renormalization strategy that generates an approximate density of states from the transition matrices of smaller subsystems.…”

“…The cluster algorithm should however only be applied at temperatures close to the critical temperature since, as noted above, away from the extent of the accessible phase space and hence the intrinsic error of the single spin flip method is far smaller. [15]…”

A projected entropy controller for transition matrix calculations

“…Another method derives from the observation that, as mentioned above, the evolution of the full E H − state distribution rather than just the distribution of states as a function of magnetization as in Ref. [1] provides an alternative metric to establish if a computation has properly converged at a given temperature. By analogy, the computed effective number of projected states together with the diffusion velocity of the Markov chain could similarly be employed to determine the number of realizations required for convergence at each temperature rather than the magnitude of the temperature step.…”

“…6 of Ref. [1]. While the accuracy is slightly less than the procedure of the reference, this is compensated by the greater simplicity of the algorithm.…”

“…Accordingly, the number of realizations at each temperature can be determined by monitoring the convergence of the normalized histogram of samples as a function of magnetization, yielding a procedure that dynamically adapts to variations in the size of the accessible projected phase space. [1] Observe that a somewhat more accurate but structurally complex technique would be to enforce convergence over the entire two-dimensional E H − distribution at each temperature.…”