We study the nature of the phase transition of lattice gauge theories at high temperature and high density by focusing on the probability distribution function, which represents the probability that a certain density will be realized in a heat bath. The probability distribution function is obtained by creating a canonical partition function fixing the number of particles from the grand partition function. However, if the $Z_3$ center symmetry, which is important for understanding the finite temperature phase transition of $SU(3)$ lattice gauge theory, is maintained on a finite lattice, the probability distribution function is always zero, except when the number of particles is a multiple of three. For $U(1)$ lattice gauge theory, this problem is more serious. The probability distribution becomes zero when the particle number is nonzero. This problem is essentially the same as the problem that the expectation value of the Polyakov loop is always zero when calculating with finite volume. In this study, we propose a solution to this problem. We also propose a method to avoid the sign problem, which is an important problem at finite density, using the center symmetry. In the case of $U(1)$ lattice gauge theory with heavy fermions, numerical simulations are actually performed, and we demonstrate that the probability distribution function at a finite density can be calculated by the method proposed in this study. Furthermore, the application of this method to QCD is discussed.

• Received 7 April 2022
• Accepted 29 November 2022

DOI:https://doi.org/10.1103/PhysRevD.106.114505

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Published by the American Physical Society