Critical behaviour of the quasi-periodic quantum Ising chain

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