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A hierarchy of self-consistent stochastic boundary conditions for Ising lattice simulations

We describe a hierarchy of stochastic boundary conditions (SBCs) that can be used to systematically eliminate finite size effects in Monte Carlo simulations of Ising lattices. For an Ising model on a $100 \times 100$ square lattice, we measured the specific heat, the magnetic susceptibility, and the spin-spin correlation using SBCs of the two lowest orders, to show that they compare favourably against periodic boundary conditions (PBC) simulations and analytical results. To demonstrate how versatile the SBCs are, we then simulated an Ising lattice with a magnetized boundary, and another with an open boundary, measuring the magnetization, magnetic susceptibility, and longitudinal and transverse spin-spin correlations as a function of distance from the boundary.