# Critical behavior of the quasi-periodic quantum Ising chain

## TL;DR

This paper investigates the critical phenomena of the quasi-periodic quantum Ising chain, revealing how spatial modulation influences universality classes, critical exponents, and excitation localization, with special cases showing unique behaviors.

## Contribution

It provides a detailed analytic and numerical analysis confirming the QP-Ising universality class and explores special phase difference cases with distinct critical properties.

## Key findings

- Logarithmic wandering coefficient governs critical exponents and localization length.
- Special phase differences lead to unique localization and critical behaviors.
- Generalized Aubry-André duality prevents localization at certain parameters.

## Abstract

The interplay of correlated spatial modulation and symmetry breaking leads to quantum critical phenomena intermediate between those of the clean and randomly disordered cases. By performing a detailed analytic and numerical case study of the quasi-periodically (QP) modulated transverse field Ising chain, we provide evidence for the conjectures of Ref. [Crowley et. al. 2018] regarding the QP-Ising universality class. In the generic case, we confirm that the logarithmic wandering coefficient $w$ governs both the macroscopic critical exponents and the energy-dependent localisation length of the critical excitations. However, for special values of the phase difference $\Delta$ between the exchange and transverse field couplings, the QP-Ising transition has different properties. For $\Delta=0$, a generalised Aubry-Andr\'e duality prevents the finite energy excitations from localising despite the presence of logarithmic wandering. For $\Delta$ such that the fields and couplings are related by a lattice shift, the wandering coefficient $w$ vanishes. Nonetheless, the presence of small couplings leads to non-trivial exponents and localised excitations. Our results add to the rich menagerie of quantum Ising transitions in the presence of spatial modulation.