Some Exact Formulas on Long-Range Correlation Functions of the Rectangular Ising Lattice

TL;DR

This paper derives exact formulas for long-range correlation functions in a rectangular Ising lattice with cyclic boundaries, revealing how correlations approach bulk values as spins are far apart.

Contribution

It provides new low-temperature series expansions and exact correlation formulas for the rectangular Ising lattice with specific boundary conditions.

Findings

Correlation functions approach product of row magnetizations at large distances

Dominant terms in series expansions increase with the row index

Exact correlation functions obtained via D log Pade method

Abstract

We study long-range correlation functions of the rectangular Ising lattice with cyclic boundary conditions. Specifically, we consider the situation in which two spins are on the same column, and at least one spin is on or near free boundaries. The low-temperature series expansions of the correlation functions are presented when the spin-spin couplings are the same in both directions. The exact correlation functions can be obtained by D log Pade for the cases with simple algebraic resultant expressions. The present results show that as the two spins are infinitely far from each other, the correlation function is equal to the product of the row magnetizations of the corresponding spins as expected. In terms of low-temperature series expansions, the approach of this m-th row correlation function to the bulk correlation function for increasing m can be understood from the observation that the dominant terms of their series expansions are the same successively in the above two correlation functions. The number of these dominant terms increases monotonically as m increases.