Program package for multicanonical simulations of U(1) lattice gauge theory

📝 Abstract

We document our Fortran 77 code for multicanonical simulations of 4D U(1) lattice gauge theory in the neighborhood of its phase transition. This includes programs and routines for canonical simulations using biased Metropolis heatbath updating and overrelaxation, determination of multicanonical weights via a Wang-Landau recursion, and multicanonical simulations with fixed weights supplemented by overrelaxation sweeps. Measurements are performed for the action, Polyakov loops and some of their structure factors. Many features of the code transcend the particular application and are expected to be useful for other lattice gauge theory models as well as for systems in statistical physics.

💡 Analysis

We document our Fortran 77 code for multicanonical simulations of 4D U(1) lattice gauge theory in the neighborhood of its phase transition. This includes programs and routines for canonical simulations using biased Metropolis heatbath updating and overrelaxation, determination of multicanonical weights via a Wang-Landau recursion, and multicanonical simulations with fixed weights supplemented by overrelaxation sweeps. Measurements are performed for the action, Polyakov loops and some of their structure factors. Many features of the code transcend the particular application and are expected to be useful for other lattice gauge theory models as well as for systems in statistical physics.

📄 Content

Continuum, quantum gauge theories are defined by their Lagrangian densities, which are functions of fields living in 4D Minkowski space-time. In the path-integral representation physical observables are averages over possible field configurations weighted with an exponential factor depending on the action. By performing a Wick rotation to imaginary time the Minkowski metric becomes Euclidean. Discretization of this 4D Euclidean space results in lattice gauge theory (LGT) -a regularization of the original continuum theory, which allows to address non-perturbative problems. For a textbook see, for instance, Ref. [1]. Physical results are recovered in the quantum continuum limit a → 0, where a is the lattice spacing measured in units proportional to a diverging correlation length.

U(1) pure gauge theory, originally introduced by Wilson [2], is a simple 4D LGT. Nevertheless, determining its phase structure beyond reasonable doubt has turned out to be a non-trivial computational task. One encounters a phase transition which is believed to be first-order on symmetric N 4 s lattices, e.g. [3,4,5]. For a finite temperature N 3 s × N τ , N τ < N s geometry the situ-ation is less clear: Either second-order for small N τ and firstorder for large N τ [6], or always second order, possibly corresponding to a novel renormalization group fixed point [7]. In 3D U(1) gauge theory is confining for all values of the coupling on symmetric N 3 s lattices [8,9], while in the finite temperature N 2 s × N τ , N τ < N s geometry a deconfining transition of Berezinsky-Kosterlitz-Thouless type [10,11] is expected, see [12] for recent numerical studies.

In lattice gauge theory one can evaluate Euclidean path integrals stochastically by generating an ensemble of field configurations with Markov chain Monte Carlo (MCMC). In this paper we report MCMC techniques we used in [7]. They are based on multicanonical (MUCA) simulations [13,14] supplemented by a Wang-Landau (WL) recursion [15], both employed in continuum formulations. For updating we use the biased Metropolis heatbath algorithm (BMHA) of [16] added by overrelaxation [17]. Observables include the specific heat, Polyakov loops and their structure factors (SFs) for low-lying momenta. For the analysis of these data binning is used to render autocorrelations negligible and a logarithmically coded reweighting procedure calculates averages with jackknife error bars.

Our program package STMC U1MUCA.tgz can be downloaded from the web at http://www.hep.fsu.edu/~berg/research . Unfolding of STMC U1MUCA.tgz with tar -zxvf creates a tree structure with root directory STMC U1MUCA. Folders on the first level are ExampleRuns, ForProg and Libs. Besides in the subfolders of ExampleRuns, copies of all main programs are found in ForProg. Fortran functions and subroutines of our code are located in the subfolders of Libs, which are Fortran, Fortran par, U1, and U1 par. Routines in Fortran and U1 are modular, so that they can be called in any Fortran program, while routines in the other two subfolders need user defined parameter files, which they include at compilation. General purpose routines are in the Fortran subfolders and to a large extent taken over from ForLib of [14], while routines specialized to U(1) are in the U1 folders. Parameter files are bmha.par, lat.par, lat.dat, mc.par

for canonical simulations and in addition sf.par, u1muca.par

for SF measurements and MUCA simulations with WL recursion. The main programs and the routines of the subfolders Fortran par and U1 par include common blocks when needed. These common blocks with names common*.f are also located in the Fortran par and U1 par subfolders. This paper is organized as follows. In Sec. 2 we define U(1)

LGT and introduce the routines for our BMHA and for measurements of some observables. Sec. 3 is devoted to our code for MUCA runs and to the analysis of these data. Sections 2 and 3 both finish with explicit example runs, where Sec. 3 uses on input action parameter estimates obtained in Sec. 2. A brief summary and conclusions are given in the final section 4.

Our code is written for a variable dimension d and supposed to work for d ≥ 2. However, its use has mainly been confined to d = 4 to which we restrict most of the subsequent discussion.

After U(1) gauge theory is discretized its fundamental degrees of freedom reside on the links of a 4D hypercubic lattice which we label by x, µ: x is a 4D vector giving the location of a site, and µ = 1, 2, 3, 4 is the direction of the link originating from this site, µ = 4 corresponds to the temporal direction of extension N τ and µ = 1, 2, 3 to the spatial directions of extension N s .

The system contains N 3 s × N τ sites and N 3 s × N τ × 4 degrees of freedom U x,µ that belong to U(1) gauge group, which we parametrize by

In our code we use Wilson’s action

The product U x,µ U x+ μ,ν U + x+ν,µ U + x,ν is taken around a plaquette, an elementary square of the lattice. In 4D each link

This content is AI-processed based on ArXiv data.