Critical behavior of the quasi-periodic quantum Ising chain

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 $Δ$ between the exchange and transverse field couplings, the QP-Ising transition has different properties. For $Δ=0$, a generalised Aubry-André duality prevents the finite energy excitations from localising despite the presence of logarithmic wandering. For $Δ$ 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.

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