On Duality of Two-dimensional Ising Model on Finite Lattice

TL;DR

This paper establishes duality relations for the 2D Ising model on finite lattices, including nonhomogeneous cases, and derives implications for correlation functions and boundary conditions.

Contribution

It generalizes duality relations for the 2D Ising model to nonhomogeneous finite lattices with various boundary conditions.

Findings

Partition function expressed via dual lattice functions.

Duality relations extended to nonhomogeneous couplings.

Derived duality relations for correlation functions and boundary conditions.

Abstract

It is shown that the partition function of the 2d Ising model on the dual finite lattice with periodical boundary conditions is expressed through some specific combination of the partition functions of the model on the torus with corresponding boundary conditions. The generalization of the duality relations for the nonhomogeneous case is given. These relations are proved for the weakly-nonhomogeneous distribution of the coupling constants for the finite lattice of arbitrary sizes. Using the duality relations for the nonhomogeneous Ising model, we obtain the duality relations for the two-point correlation function on the torus, the 2d Ising model with magnetic fields applied to the boundaries and the 2d Ising model with free, fixed and mixed boundary conditions.