Critical Behavior of Hierarchical Ising Models

TL;DR

This paper investigates the critical behavior of layered two-dimensional Ising models with hierarchical couplings, revealing how non-periodic modulations influence critical exponents and anisotropic divergence near phase transitions.

Contribution

It provides a detailed analytical and numerical analysis of how hierarchical, non-periodic couplings affect critical phenomena in both classical and quantum Ising models.

Findings

Critical exponents vary continuously with modulation strength.

Near criticality, the system exhibits anisotropic divergence of correlation lengths.

The nature of the perturbation (irrelevant, relevant, marginal) influences critical behavior.

Abstract

We consider the critical behavior of two-dimensional layered Ising models where the exchange couplings between neighboring layers follow hierarchical sequences. The perturbation caused by the non-periodicity could be irrelevant, relevant or marginal. For marginal sequences we have performed a detailed study, which involved analytical and numerical calculations of different surface and bulk critical quantities in the two-dimensional classical as well as in the one-dimensional quantum version of the model. The critical exponents are found to vary continuously with the strength of the modulation, while close to the critical point the system is essentially anisotropic: the correlation length is diverging with different exponents along and perpendicular to the layers.