Iterated star-triangle transformation on inhomogeneous 2D Ising lattices

TL;DR

This paper investigates the iterative star-triangle transformation on inhomogeneous 2D Ising lattices, revealing that certain transformed variables remain confined to specific complex planes, thus providing a foundation for analyzing frustrated Ising systems.

Contribution

It introduces a framework for analyzing the iterative star-triangle transformation on inhomogeneous 2D Ising lattices, including frustrated cases, and shows the confinement of transformed variables to specific complex domains.

Findings

Variables $1/ anh 2K_i^{(n)}$ stay on the union of real and imaginary axes.

The confinement holds for infinite, periodic, and finite lattices with boundary protocols.

The study provides a basis for future analytic and numerical analysis of frustrated Ising models.

Abstract

We consider infinite or periodic 2D triangular Ising lattices with arbitrary positive or negative nearest-neighbor couplings $K_i(\vec{r})$, where $\vec{r}$ and $i$ indicate the bond position and orientation, respectively. Iterative application of the star-triangle transformation to an initial lattice $\mathcal{T}(0)$ with a set of couplings $\{K_i^{(0)}(\vec{r})\}$ generates a sequence of lattices $\mathcal{T}(n)$, for $n=1,2,\ldots,$ with couplings $\{K_i^{(n)}(\vec{r})\}$. When $\mathcal{T}(0)$ includes sufficiently strongly frustrated plaquettes, complex couplings will appear. We show that, nevertheless, the variables $1/\sinh 2K_i^{(n)}\!(\vec{r})$ remain confined to the union ${\mathbb{R}} \cup \rm{i}{\mathbb{R}}$ of the real and the imaginary axis. The same holds for a lattice with free boundaries, provided we distinguish between "receding" and "advancing" boundaries, the latter having degrees of freedom that must be fixed by an appropriately chosen protocol. This study establishes a framework for future analytic and numerical work on such frustrated Ising lattices.