Extension of the constant exchange probability method to multi-dimensional replica exchange Monte Carlo applied to the tri-critical spin-1 Blume–Capel model

In replica exchange Monte Carlo (REM), tuning of the temperature set and the exchange scheduling are crucial in improving the accuracy and reducing calculation time. In multi-dimensional simulated tempering, the first order phase transition is accessible. Therefore it is important to study the tuning of parameter set and the scheduling of exchanges in the parallel counterpart, the multi-dimensional REM. We extend Hukushima's constant exchange probability method to multi-dimensional REM for the parameter set. We further propose a combined method to use this set and the Bittner-Nussbaumer-Janke's PT tau algorithm for scheduling. We test the proposed method in two-dimensional spin-1 Blume-Capel model and find that it works efficiently, including the vicinity of the first order phase transition.