Generalized Transformation for Decorated Spin Models

TL;DR

This paper introduces a generalized analytical transformation that converts decorated Ising models with complex interactions into simpler undecorated models, facilitating the calculation of partition functions and correlation functions.

Contribution

It extends existing transformation methods to include long-range and high-order interactions, providing a unified approach for various decorated spin systems.

Findings

Transformation applies to mixed spin square lattices.

Effective models include multi-body interactions.

Method simplifies analysis of complex decorated systems.

Abstract

The paper discusses the transformation of decorated Ising models into an effective \textit{undecorated} spin models, using the most general Hamiltonian for interacting Ising models including a long range and high order interactions. The inverse of a Vandermonde matrix with equidistant nodes $[-s,s]$ is used to obtain an analytical expression of the transformation. This kind of transformation is very useful to obtain the partition function of decorated systems. The method presented by Fisher is also extended, in order to obtain the correlation functions of the decorated Ising models transforming into an effective undecorated Ising models. We apply this transformation to a particular mixed spin-(1/2,1) and (1/2,2) square lattice with only nearest site interaction. This model could be transformed into an effective uniform spin-$S$ square lattice with nearest and next-nearest interaction, furthermore the effective Hamiltonian also include combinations of three-body and four-body interactions, particularly we considered spin 1 and 2.