https://doi.org/10.1140/epjc/s10052-016-4142-5
Regular Article - Theoretical Physics
An efficient algorithm for numerical computations of continuous densities of states
1
Centre for Mathematical Sciences, Plymouth University, Plymouth, PL4 8AA, UK
2
College of Science, Swansea University, Singleton Park, Swansea, SA2 8PP, UK
3
School of Physics and Astronomy, University of Edinburgh, Edinburgh, EH9 3FD, UK
* e-mail: b.lucini@swansea.ac.uk
Received:
15
February
2016
Accepted:
16
May
2016
Published online:
2
June
2016
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 
to 
. Our results perfectly agree with the reference values reported in the literature, which covers lattice sizes up to 
. Robust results for the 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.
© The Author(s), 2016
