Exact Renormalization-Group Study of Aperiodic Ising Quantum Chains and Directed Walks

TL;DR

This paper uses exact renormalization-group methods to analyze the critical behavior of aperiodic Ising quantum chains and directed walks, revealing rich nonuniversal critical phenomena and their scaling relations.

Contribution

It establishes a precise connection between the spectral properties of transfer matrices and the thermodynamics of aperiodic Ising and directed walk models, providing exact results for their critical behavior.

Findings

Nonuniversal critical exponents depend on coupling distributions.

Rich bulk and surface critical behaviors including first-order transitions.

Scaling relations link critical exponents across models.

Abstract

We consider the Ising model and the directed walk on two-dimensional layered lattices and show that the two problems are inherently related: The zero-field thermodynamical properties of the Ising model are contained in the spectrum of the transfer matrix of the directed walk. The critical properties of the two models are connected to the scaling behavior of the eigenvalue spectrum of the transfer matrix which is studied exactly through renormalization for different self-similar distributions of the couplings. The models show very rich bulk and surface critical behaviors with nonuniversal critical exponents, coupling-dependent anisotropic scaling, first-order surface transition, and stretched exponential critical correlations. It is shown that all the nonuniversal critical exponents obtained for the aperiodic Ising models satisfy scaling relations and can be expressed as functions of varying surface magnetic exponents.