An efficient algorithm for numerical computations of continuous densities of states

Abstract

In Wang-Landau type algorithms, Monte-Carlo updates are performed with respect to the density of states, which is iteratively refined during simulations. The partition function and thermodynamic observables are then obtained by standard integration. In this work, our recently introduced method in this class (the LLR approach) is analysed and further developed. Our approach is a histogram free method particularly suited for systems with continuous degrees of freedom giving rise to a continuum density of states, as it is commonly found in Lattice Gauge Theories and in some Statistical Mechanics systems. We show that the method possesses an exponential error suppression that allows us to estimate the density of states over several orders of magnitude with nearly-constant relative precision. We explain how ergodicity issues can be avoided and how expectation values of arbitrary observables can be obtained within this framework. We then demonstrate the method using Compact U(1) Lattice Gauge Theory as a show case. A thorough study of the algorithm parameter dependence of the results is performed and compared with the analytically expected behaviour. We obtain high precision values for the critical coupling for the phase transition and for the peak value of the specific heat for lattice sizes ranging from 8 4 to 20 4 . Our results perfectly agree with the reference values reported in the literature, which covers lattice sizes up to 18 4 . Robust results for the 20 4 volume are obtained for the first time. This latter investigation, which, due to strong metastabilities developed at the pseudo-critical coupling of the system, so far has been out of reach even on supercomputers with importance sampling approaches, has been performed to high accuracy with modest computational resources. This shows the potential of the method for studies of first order phase transitions. Other situations where the method is expected to be superior to importance sampling techniques are pointed out.

Controlling the sign problem in finite-density quantum field theory

“…The appeal of method (i) is that it is universally applicable if a way is found to control the cancellations. A first success for direction (i) emerged with the advent of Wang-Landau type techniques and, most notably, the LLR method [11]: due to the feature of exponential error suppression of the LLR approach [12], high precision data for the density of states ρ(s) of finding a particular phase s over many orders of magnitude has become available. The partition function now emerges as Fourier transform of ρ(s).…”

Density of states techniques for lattice field theories using the functional fit approach (FFA)

“…To reduce the effect of the statistical fluctuations it has been proposed [3] to fit the density ρ(x) with a polynomial in x, which can be chosen to be an even polynomial here, since ρ(x) is even. We implemented this suggestion in our analysis of the results for M − M * and indeed we find a drastic improvement of the agreement with the reference data from the dual simulation (see [9] for the details).…”

“…The determination of the parameters k n of ρ(x) was implemented as described in Section 2. For a similar implementation of a DoS analysis of 4-dimensional U(1) lattice gauge theory with the DoS LLR approach see the last reference in [3].…”

Density of states

“…To my knowledge, the first of its kind is the Multicanonical Algorithm by Berg and Neuhaus [9]. The same re-weighting approach is also at the heart of the Wang-Landau approach [10] or, more recently a version adapted to continuous QFT degrees of freedom, the LLR method [11,12], which will be the focal point of this paper (also see [13] for a recent review).…”