Evaluation of an Efficient Monte Carlo Algorithm to Calculate the Density of States

Abstract

Phase transitions and critical phenomena are the most universal phenomena in nature. To understand the phase transitions and critical phenomena of a given system as a continuous function of temperature and to obtain the partition function zeros in the complex temperature plane indicating most effectively phase transitions and critical phenomena, we need to calculate the density of states. Currently, Wang-Landau Monte Carlo algorithm is one of the most efficient Monte Carlo methods to calculate the approximate density of states. Using Wang-Landau Monte Carlo algorithm, the density of states for the Ising model on L x L square lattices (L = 4 ~ 32) with periodic boundary conditions is obtained, and the partition function zeros of the Ising model are evaluated in the complex temperature plane. By examining the behavior of the first partition function zero (partition function zero closest to the positive real axis), phase transitions and critical phenomena can be much more accurately analyzed. The approximate first zeros of the Ising ferromagnet, obtained from Wang-Landau algorithm, are quite close to the exact ones, indicating that it is a reliable method for calculating the density of states and the first partition function zeros. function zeros indicating most effectively phase transitions and critical phenomena, and to perform microcanonical analysis of phase transitions and critical phenomena, we need to calculate the density of states. Currently, Wang-Landau Monte Carlo algorithm (3) is one of the most efficient Monte Carlo methods to calculate the approximate density of states. In Wang-Landau Monte Carlo algorithm, the inverse of the density of states is employed as the sampling probability function, and the real values for the density of states can be obtained quickly due to its modification factor. Phase transitions and critical phenomena can also be understood based on the concept of partition function zeros. Fisher introduced the partition function zeros in the complex temperature plane utilizing the Onsager solution of the square-lattice Ising model in the absence of an external magnetic field (4). Fisher also showed that the partition function zeros in the complex temperature plane of the square-lattice Ising model determine its ferromagnetic and antiferromagnetic critical temperatures at the same time in the absence an external magnetic field. By calculating the partition function zeros and examining the behavior of the first partition function zero (partition function zero closest to the positive real axis) in the thermodynamic limit, phase transitions and critical phenomena can be much more accurately analyzed than examining the behavior of the specific heat per volume for real values of the temperature, which is plagued by the noise due to the subleading terms containing zeros other than the first ones (5−20). In the next section, the density of states and the partition function of the square-lattice Ising model are defined. In Section III, Wang-Landau Monte Carlo algorithm to calculate the approximate density of states is briefly explained. In Section IV, the concept of the partition function zeros in the complex temperature plane is introduced, the partition function zeros of the square-lattice Ising model are evaluated in the complex temperature plane using the exact density of states and the approximate density of states, and both results are compared.