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.
Highlights
There are noticeable cases in which Monte-Carlo importance sampling methods are either very inefficient or produce inherently wrong results for well understood reasons
The main purpose of this work is to discuss in detail some improvements of the original logarithmic linear relaxation (LLR) algorithm and to formally prove that expectation values of observables computed with this method converge to the correct result, which fills a gap in the current literature
We apply the algorithm to the study of compact U(1) lattice gauge theory, a system with severe metastabilities at its first order phase transition point that make the determination of observables near the transition very difficult from a numerical point of view
Summary
Owing to formal similarities between the two fields, the approach we are proposing can be applied to both statistical mechanics and lattice field theory systems. Z (β) = Dφ eβ S[φ], which defines the canonical partition function for the statistical system or the path integral in the field theory case. The density of state (which is a function of the value of S[φ] = E) is formally defined by the integral. A numerical determination of ρ(E) would enable us to express Z and O as numerical integrals of known functions in the single variable E. This approach is inherently different from conventional Monte-Carlo calculations, which relies on the concept of importance sampling, i.e. the configurations contributing to the integral are generated with probability. Owing to this conceptual difference, the method we are proposing can overcome notorious drawbacks of importance sampling techniques
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