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Abstract

Phase transitions are widely studied in the context of statistical physics and condensed matter systems, but the principles of this study can be extended to more generalized field theories. We examine a 1+1 U(1) quantum field theory in one spatial and one Euclidean temporal dimension that displays a dualism with the classical rotor model in 2D. The classical rotor model is known for the BKT (Berezinskii–Kosterlitz–Thouless) transition which is not consistent with the Landau paradigm of phase transitions and is instead driven by the proliferation of vortex field solutions. We use this dualism to examine a Villain formulation of our U(1) QFT and observe the analogous phase transition.

To probe the phase transition we use Monte Carlo methods to calculate expectation values in the path integral formulation of QFT via a novel Python-based software package. We begin by using this package to reproduce supercomputer mappings of the BKT transition performed in 1993, which yielded a transition point at a critical thermodynamic ß~0.74. In the latter part of this paper, we develop the so-called Worldline formulation, dual to the Villain formulation via Poisson resummation, which allows the use of "worm" algorithms for Monte Carlo updates. These worm algorithms can circumvent the "critical slowing" of the Monte Carlo Markov Chain autocorrelation, which is otherwise characteristic of such studies. Further applications, including the implementation of vortex winding constraints, are discussed.

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