Bivariate Transition Matrix Monte Carlo Method for Joint Density of States Calculations

While most thermodynamic variables of a statistical system can be evaluated at all temperatures from the density of states, if a phase transition is present, quantities such as Landau free energy and the probability distribution of the order parameter must instead be determined from the joint density of states which is a function of both the energy and a second variable, typically the order parameter.
This talk demonstrates that by combining the transition matrix Monte Carlo method with the bivariate multicanonical sampling the joint density of states can be efficiently and accurately computed. The procedure is then applied to the Ising and Potts models as well as to Ising spin glasses. The Landau free energies, the probability distribution of the order parameter and the Binder cumulants are calculated. Finally, we discuss the implications of our results with regard to the existence of a nonzero temperature phase transition in the two-dimensional Ising spin glass.