Surface Magnetization and Critical Behavior of Aperiodic Ising Quantum Chains

TL;DR

This paper investigates how different aperiodic modulations affect the surface magnetization and critical behavior of two-dimensional layered Ising models, revealing varied critical phenomena depending on the type of sequence.

Contribution

It derives exact functional equations for surface magnetization in aperiodic Ising models and analyzes their critical behavior for three specific sequences, including new anomalous surface critical phenomena.

Findings

Thue-Morse sequence shows irrelevant perturbation with square root singularity.

Period-doubling sequence exhibits marginal perturbation with continuously varying exponents.

Rudin-Shapiro sequence demonstrates relevant perturbation with either essential singularity or first-order transition.

Abstract

We consider semi-infinite two-dimensional layered Ising models in the extreme anisotropic limit with an aperiodic modulation of the couplings. Using substitution rules to generate the aperiodic sequences, we derive functional equations for the surface magnetization. These equations are solved by iteration and the surface magnetic exponent can be determined exactly. The method is applied to three specific aperiodic sequences, which represent different types of perturbation, according to a relevance-irrelevance criterion. On the Thue-Morse lattice, for which the modulation is an irrelevant perturbation, the surface magnetization vanishes with a square root singularity, like in the homogeneous lattice. For the period-doubling sequence, the perturbation is marginal and the surface magnetic exponent varies continuously with the modulation amplitude. Finally, the Rudin-Shapiro sequence, which corresponds to the relevant case, displays an anomalous surface critical behavior which is analyzed via scaling considerations: Depending on the value of the modulation, the surface magnetization either vanishes with an essential singularity or remains finite at the bulk critical point, i.e., the surface phase transition is of first order.