Performance limitations of flat-histogram methods.

TLDR: The shape parameters of these distributions indicate that statistical sample means become ill defined already for moderate system sizes within these complex energy landscapes, as well as indicating the optimal scaling of local-update flat-histogram methods with system size.

Abstract: Monte Carlo methods are well-suited for the simulation of large many body problems, since the complexity for a single Monte Carlo update step scales only polynomially and often linearly in the system size, while the config- uration space grows exponentially with the system size. The performance of a Monte Carlo method is then deter- mined by how many update steps are needed to efficiently sample the configuration space. For second order phase transitions in unfrustrated systems the problem of "crit- ical slowing down" - a rapid divergence of the number of Monte Carlo steps needed to obtain a subsequent un- correlated configuration - was solved more than a decade ago by cluster update algorithms (1). At first order phase transitions and in systems with many local minima of the free energy such as frustrated magnets or spin glasses, there is the similar problem of long tunneling times be- tween local minima. With energy barriersE scaling lin- early with the linear system size L, the tunneling times � at an inverse temperature � = 1/kBT scale exponentially with the system size, � � exp(��E) / exp(const × L). Several methods were developed to overcome this tun- neling problem, such as the multicanonical method (2), broad histograms (4), simulated and parallel tempering (3), and Wang-Landau sampling (5). The common aim of all these methods is to broaden the range of energies sam- pled within Monte Carlo simulations from the sharply peaked distribution of canonical sampling at fixed tem- perature in order to ease the tunneling through barriers. Ideally, all relevant energy levels are sampled equally often during a simulation, thus producing a "flat his- togram" in energy space. Some methods approach this goal by variations and generalizations of canonical dis- tributions (2, 3), while others (4, 5) discard the notion of temperature completely and instead are formulated in terms of the density of states. With a probability p(E) for a single configuration with energy E, the probability of sampling an arbitrary configuration with energy E is given as PE = �(E)p(E), where the density of states �(E) counts the number of states with energy E. Upon choos- ing p(E) / 1/�(E) instead of p(E) / exp( �E) one ob- tains a constant probability PE for visiting each energy level E, and hence a flat histogram. Wang and Landau (5) proposed a simple and elegant flat histogram algorithm that iteratively improves approximations to the initially unknown density of states �(E). Once �(E) is determined with sufficient accuracy, the Monte Carlo algorithm just performs a random walk in energy space. Within two years of publication this algorithm has been applied to a large number of problems (6, 7, 8) and extended to quantum systems (9). In this Letter we investigate the performance of flat histogram algorithms in general, and the Wang-Landau algorithm in particular, for three systems for which the density of states �(E) is known exactly on finite two- dimensional (2D) lattices: the Ising ferromagnet as the simplest example, the fully frustrated Ising model as a prototype for frustrated systems, and the ±J Ising spin glass. For each of these models we construct a perfect flat histogram method by simulating a random walk in configuration space where we employ the known density of states for these models to set p(E) / 1/�(E). As a measure of performance we use the average tun- neling timeto get from a ground state (lowest energy configuration) to an anti-ground state (configuration of highest energy), which is the relevant time scale for sam- pling the whole phase space (10). Since the number of energy levels in a d-dimensional system with linear size L scales with the number of spins N = L d , the tunneling time for a pure random walk in energy space is