Ising Problem on Simple Cubic Lattice

TL;DR

This paper establishes a novel correspondence between simple cubic and plane triangular lattices to solve the 3D Ising problem and introduces a new high-temperature expansion method that avoids explicit graph counting.

Contribution

It presents an exact correspondence between SC and PT lattices and proposes a new high-temperature expansion method applicable to 2D and 3D lattices that simplifies calculations.

Findings

Exact one-to-one correspondence between closed graphs on SC and PT lattices

A new high-temperature expansion method that avoids explicit graph counting

Potential to achieve exact series expansions to high order

Abstract

Simple cubic lattice (SC lattice) can be viewed as plane triangular lattice (PT lattice) by viewing it along its principle diagonal lines. By viewing thus we establish the exact one-to-one correspondence between the closed graphs on SC lattice and the corresponding closed graphs on PT lattice. We thus see that the propagator for PT lattice (with suitable modifications) can be used to solve, at least in principle, the 3D Ising problem for SC lattice in the absence of external magnetic field. A new method is then proposed to generate high temperature expansion for the partition function. This method is applicable to 2D as well as 3D lattices. This method does not require explicit counting of closed graphs and this counting is achieved in an indirect way and thus exact series expansion can be achieved up to any sufficiently large order.