Equivalence between two-dimensional alternating/random Ising model and the ground state of one-dimensional alternating/random XY chain

TL;DR

This paper establishes an exact equivalence between a two-dimensional alternating/random Ising model and the ground state of a one-dimensional alternating/random XY chain, linking two important models in statistical physics.

Contribution

It demonstrates a precise equivalence between 2D Ising models with randomness and 1D XY chains, extending understanding of their phase behaviors.

Findings

Equivalence holds for models with periodic and free boundary conditions.

The random Ising model with Griffiths-McCoy phase corresponds to a random XY chain.

Provides a new perspective for analyzing complex disordered systems.

Abstract

It is derived that the two-dimensional Ising model with alternating/random interactions and with periodic/free boundary conditions is equivalent to the ground state of the one-dimensional alternating/random XY model with the corresponding periodic/free boundary conditions. This provides an exact equivalence between a random rectangular Ising model, in which the Griffiths-McCoy phase appears, and a random XY chain.