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Description
We investigate the phase structure and equations of state in lattice gauge theory at finite temperature and finite density, focusing on the canonical partition function obtained by expanding the grand partition function in terms of particle number. We demonstrate that when quark masses are heavy and the hopping parameter expansion of the quark determinant is applicable, the complex phase of the quark determinant, which causes the sign problem, can be controlled by decomposing the grand partition function according to particle number. In this study, we propose an algorithm to calculate the canonical partition function for heavy-quark, high-density effective theory and first tested it using SU(2) gauge theory with one-flavor, a case where the sign problem arises. When the grand partition function of SU(2) is decomposed into canonical partition functions, it can be seen that the sign problem is serious in the part with an odd number of particles, while it is not so serious in the part with an even number of particles. Furthermore, the odd-particle sectors present a fundamental problem. Due to the center symmetry of the theory, the canonical partition function of SU(N) gauge theory vanishes unless the particle number is an integer multiple of N. While this is valid in the confinement phase, it is physically incorrect in the deconfinement phase. We resolve this issue. We perform Monte Carlo simulations, calculate the canonical partition function without the sign problem, and reconstruct the grand partition function. Then, we compute the equation of state for thermodynamic quantities and the behavior of order parameters across the entire range of temperature and density.