The quasi-periodic quantum Ising transition in 1D

TL;DR

This paper investigates how strong quasi-periodic modulation in a 1D quantum Ising chain induces localized, gapless phases and alters the critical behavior, revealing a new universality class influenced by local coupling wandering.

Contribution

It demonstrates that strong quasi-periodic modulation leads to localized, gapless phases and introduces a new universality class controlled by the wandering coefficient.

Findings

Strong modulation induces localized, gapless phases.

The critical exponents differ from clean and disordered models.

Wandering coefficient $w$ influences the universality class.

Abstract

Unlike random potentials, quasi-periodic modulation can induce localisation-delocalisation transitions in one dimension. In this article, we analyse the implications of this for symmetry breaking in the quasi-periodically modulated quantum Ising chain. Although weak modulation is irrelevant, strong modulation induces new ferromagnetic and paramagnetic phases which are fully localised and gapless. The quasi-periodic potential and localised excitations lead to quantum criticality that is intermediate to that of the clean and randomly disordered models with exponents of $\nu=1^{+}$, and $z\approx 1.9$, $\Delta_\sigma \approx 0.16$, $\Delta_\gamma\approx 0.63$ (up to logarithmic corrections). Technically, the clean Ising transition is destabilized by logarithmic wandering of the local reduced couplings. We conjecture that the wandering coefficient $w$ controls the universality class of the quasi-periodic transition and show its stability to smooth perturbations that preserve the quasi-periodic structure of the model.