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 sp
ecific 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.
Given the ground state wavefunction for an interacting lattice model, we define a correlation density matrix(CDM) for two disjoint, separated clusters $A$ and $B$, to be the density matrix of their union, minus the direct product of their respective
density matrices. The CDM can be decomposed systematically by a numerical singular value decomposition, to provide a systematic and unbiased way to identify the operator(s) dominating the correlations, even unexpected ones.
