  We performed two-dimensional simulated tempering (ST) simulations of the
two-dimensional Ising model with different lattice sizes in order to
investigate the two-dimensional ST's applicability to dealing with phase
transitions and to study the crossover of critical scaling behavior. The
external field, as well as the temperature, was treated as a dynamical variable
updated during the simulations. Thus, this simulation can be referred to as
"Simulated Tempering and Magnetizing (STM)." We also performed the "Simulated
Magnetizing" (SM) simulations, in which the external field was considered as a
dynamical variable and temperature was not. As has been discussed by previous
studies, the ST method is not always compatible with first-order phase
transitions. This is also true in the magnetizing process. Flipping of the
entire magnetization did not occur in the SM simulations under $T_\mathrm{c}$
in large lattice-size simulations. However, the phase changed through the high
temperature region in the STM simulations. Thus, the dimensional extension let
us eliminate the difficulty of the first-order phase transitions and study wide
area of the phase space. We then discuss how frequently parameter-updating
attempts should be made for optimal convergence. The results favor frequent
attempts. We finally study the crossover behavior of the phase transitions with
respect to the temperature and external field. The crossover behavior was
clearly observed in the simulations in agreement with the theoretical
implications.
