Optimized broad-histogram ensembles for the simulation of quantum systems

Abstract

The efficiency of statistical sampling in broad-histogram Monte Carlo simulations can be considerably improved by optimizing the simulated extended ensemble for fastest equilibration. Here we describe how a recently developed feedback algorithm can be generalized to find optimized sampling distributions for the simulation of quantum systems in the context of the stochastic series expansion (SSE) when defining an extended ensemble in the expansion order. If the chosen update method is efficient, such as non-local updates for systems undergoing a second-order phase transition, the optimized ensemble is characterized by a flat histogram in the expansion order if a variable-length formulation of the SSE is used. Whenever the update method suffers from slowdown, such as at a first-order phase transition, the feedback algorithm shifts weight towards the expansion orders in the transition region, resulting in a non-uniform histogram.

Classical and quantum anisotropic Heisenberg antiferromagnets

“…Because we do not find any evidence for BC structures in our simulations of the quantum model, there seems to be no mechanism for destroying long-range order at low temperatures in the quantum case. Thence the direct transition between the AF and SF phases seems to be possible, in accordance with recent quantum MC findings [20]. In general, quantum fluctuations may substantially reduce BC structures, compared to the classical case, as will be discussed below.…”

Critical entropy of quantum Heisenberg magnets on simple-cubic lattices

“…In order to extract the temperature dependence of the entropy, we use an optimized extended ensemble approach 17,18 , that is based on a generalization of the Wang-Landau 19 algorithm to the case of quantum Monte Carlo simulations 16,20 , performed within the stochastic series expansion representation 21 using multi-cluster deterministic loop updates 22 . Within this approach, one obtains Monte Carlo estimates of the expansion coefficients g(n) of the high-temperature series expansion of the partition function Z in the inverse temperature β = 1/(k B T ),…”

Measuring von Neumann entanglement entropies without wave functions

“…This can be achieved using QMC algorithms based on Wang-Landau (WL) sampling [38]. Below, we illustrate this by applying the quantum version of the WL method performed in the stochastic series expansion (SSE) QMC framework [39,40]. Compared with the conventional quantum Monte Carlo (QMC) simulations, that is performed at a fixed temperature, the WL method features two main advantages for the study of the thermodynamic properties of the EH: (i) it allows to directly compute the thermal entropy at the 'entanglement temperature' β EH , and (ii) the thermodynamic properties of the EH are obtained for a broad range of temperature with a single run of the simulation.…”

“…The use of loop updates is particularly important to avoid problems with the slowing down of the configuration selection process in a inhomogeneous Hamiltonian such as the BW-EH [15] and at critical points [47]. We refer to [39,40] for the general details of the computation of g(n), and in the appendix we discuss the technical aspects of the simulation that are relevant to reproduce our results.…”