Finite size scaling analysis of Ising model block distribution functions

TLDR: In this article, the authors studied the distribution function P L ( s ) of the local order parameters in finite blocks of linear dimension L for Ising lattices of dimension d = 2,3 and 4.

Abstract: The distribution function P L ( s ) of the local order parameters in finite blocks of linear dimension L is studied for Ising lattices of dimensionality d = 2,3 and 4. Apart from the case where the block is a subsystem of an infinite lattice, also the distribution in finite systems with free [ P L ( f ) ( s )] and periodic [ P L ( p ) ( s )] boundary conditions is treated. Above the critical point T c , these distributions tend for large L towards the same gaussian distribution centered around zero block magnetization, while below T c these distributions tend towards two gaussians centered at ± M , where M is the spontaneous magnetization appearing in the infinite systems. However, below T c the wings of the distribution at small | s | are distinctly nongaussian, reflecting two-phase coexistence. Hence the distribution functions can be used to obtain the interface tension between ordered phases. At criticality, the distribution functions tend for large L towards scaled universal forms, though dependent on the boundary conditions. These scaling functions are estimated from Monte Carlo simulations. For subsystem-blocks, good agreement with previous renormalization group work of Bruce is obtained. As an application, it is shown that Monte Carlo studies of critical phenomena can be improved in several ways using these distribution functions: ( i ) standard estimates of order parameter, susceptibility, interface tension are improved ( ii ) T c can be estimated independent of critical exponent estimates ( iii ) A Monte Carlo “renormalization group” similar to Nightingale's phenomenological renormalization is proposed, which yields fairly accurate exponent estimates with rather moderate effort ( iv ) Information on coarse-grained hamiltonians can be gained, which is particularly interesting if the method is extended to more general Hamiltonians.