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We performed two-dimensional simulated tempering (ST) simulations of thetwo-dimensional Ising model with different lattice sizes in order toinvestigate the two-dimensional ST's applicability to dealing with phasetransitions and to study the crossover of critical scaling behavior. Theexternal field, as well as the temperature, was treated as a dynamical variableupdated during the simulations. Thus, this simulation can be referred to as"Simulated Tempering and Magnetizing (STM)." We also performed the "SimulatedMagnetizing" (SM) simulations, in which the external field was considered as adynamical variable and temperature was not. As has been discussed by previousstudies, the ST method is not always compatible with first-order phasetransitions. This is also true in the magnetizing process. Flipping of theentire magnetization did not occur in the SM simulations under $T_\mathrm{c}$in large lattice-size simulations. However, the phase changed through the hightemperature region in the STM simulations. Thus, the dimensional extension letus eliminate the difficulty of the first-order phase transitions and study widearea of the phase space. We then discuss how frequently parameter-updatingattempts should be made for optimal convergence. The results favor frequentattempts. We finally study the crossover behavior of the phase transitions withrespect to the temperature and external field. The crossover behavior wasclearly observed in the simulations in agreement with the theoreticalimplications.

Simulated Tempering and Magnetizing: An Application of Two-Dimensional Simulated Tempering to Two-Dimensional Ising Model and Its Crossover