Monte Carlo studies of finite-size scaling in the Gibbs ensemble at criticality for the two-dimensional Ising model.

Abstract

In this paper we study finite-size scaling at criticality in the Gibbs ensemble with Monte Carlo simulations. A recent lattice-gas version of Gibbs-ensemble sampling for the two-dimensional Ising-model magnet with nearest-neighbor ferromagnetic interactions on two subsystems of L\ifmmode\times\else\texttimes\fi{}L lattices is considered. Both square and triangular lattices with a range of lattice sizes are simulated at bulk criticality. We are interested in the question of universality for the fourth-order cumulant of the order-parameter distribution in the Gibbs ensemble. Estimates of -1.954\ifmmode\pm\else\textpm\fi{}0.005 and -1.9543\ifmmode\pm\else\textpm\fi{}0.005 are obtained for the square and triangular lattices, respectively. These values confirm universality within the Gibbs ensemble, but are statistically different from the canonical-ensemble values of \ensuremath{\sim}-1.83. We point out that one must interpret this apparent ensemble dependence with care. Unlike in the canonical ensemble, the order parameter in the Gibbs ensemble is the subsystem magnetization. Its fourth-order cumulant is not simply related to the usual fourth magnetic-field derivative of the total free energy. Standard finite-size scaling analysis of data for the total specific heat, subsystem magnetization, and susceptibility are also considered and found to be in good agreement with finite-size scaling.