Aperiodic Ising Models

TL;DR

This paper explores the properties of non-periodic Ising models with intrinsic long-range aperiodic order, analyzing their spectral and critical behaviors in one and two dimensions using substitution sequences and exact solutions.

Contribution

It introduces a detailed study of aperiodic Ising models with substitution sequence modulations and analyzes their spectral properties and critical behavior using duality and transfer matrix methods.

Findings

Lee-Yang zeros exhibit characteristic gap structures following the gap labeling theorem.

Fermion frequencies in quantum models show similar gap structures.

Exact solutions provide insights into critical behavior on aperiodic graphs.

Abstract

We consider several aspects of non-periodic Ising models in one and two dimensions. Here we are not interested in random systems, but rather in models with intrinsic long-range aperiodic order. The most prominent examples in one dimension are sequences generated by substitution rules on a finite alphabet. The classical one-dimensional Ising chain as well as the Ising quantum chain with coupling constants modulated according to substitution sequences are considered. Both the distribution of Lee-Yang zeros on the unit circle in the classical case and that of fermion frequencies in the quantum model show characteristic gap structures which follow the gap labeling theorem of Bellissard. We also investigate the zero-field Ising model on two-dimensional aperiodic graphs, which are constructed from rectangular grids in the same spirit as the so-called Labyrinth. Here, duality arguments and exact solutions in the sense of commuting transfer matrices are used to gain information about the critical behaviour.