Anisotropic Heisenberg model on hierarchical lattices with aperiodic interactions: a renormalization-group approach

TL;DR

This study uses a real-space renormalization-group method to analyze the phase diagrams of anisotropic quantum Heisenberg models on hierarchical lattices with aperiodic interactions, confirming predictions of the Harris-Luck criterion.

Contribution

It applies a renormalization-group approach to anisotropic Heisenberg models with aperiodic interactions, revealing phase diagram types and confirming the Harris-Luck relevance criterion.

Findings

The isotropic Heisenberg plane remains invariant under the RG flow.

Critical behavior is governed by fixed points on the Ising model plane.

Relevance of aperiodic interactions aligns with the Harris-Luck criterion.

Abstract

Using a real-space renormalization-group approximation, we study the anisotropic quantum Heisenberg model on hierarchical lattices, with interactions following aperiodic sequences. Three different sequences are considered, with relevant and irrelevant fluctuations, according to the Luck-Harris criterion. The phase diagram is discussed as a function of the anisotropy parameter $\Delta$ (such that $\Delta=0$ and $\Delta=1$ correspond to the isotropic Heisenberg and Ising models, respectively). We find three different types of phase diagrams, with general characteristics: the isotropic Heisenberg plane is always an invariant one (as expected by symmetry arguments) and the critical behavior of the anisotropic Heisenberg model is governed by fixed points on the Ising-model plane. Our results for the isotropic Heisenberg model show that the relevance or irrelevance of aperiodic models, when compared to their uniform counterpart, is as predicted by the Harris-Luck criterion. A low-temperature renormalization-group procedure was applied to the \textit{classical} isotropic Heisenberg model in two-dimensional hierarchical lattices: the relevance criterion is obtained, again in accordance with the Harris-Luck criterion.