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arXiv:1801.07269 · Physical Review Letters

Quasiperiodic Quantum Ising Transitions in 1D

Philip J. D. Crowley, Anushya Chandran, Chris R. Laumann

Published Apr 24, 2018• 32 citations
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Abstract

Unlike random potentials, quasiperiodic modulation can induce localization-delocalization transitions in one dimension. In this Letter, we analyze the implications of this for symmetry breaking in the quasiperiodically modulated quantum Ising chain. Although weak modulation is irrelevant, strong modulation induces new ferromagnetic and paramagnetic phases which are fully localized and gapless. The quasiperiodic potential and localized excitations lead to quantum criticality that is intermediate to that of the clean and randomly disordered models with exponents of ν=1^{+} (exact) and z≈1.9, Δ_{σ}≈0.16, and Δ_{γ}≈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 quasiperiodic transition and show its stability to smooth perturbations that preserve the quasiperiodic structure of the model.

DOI
10.1103/physrevlett.120.175702

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Unlike random potentials, quasiperiodic modulation can induce localization-delocalization transitions in one dimension. In this Letter, we analyze the implications of this for symmetry breaking in the quasiperiodically modulated quantum Ising chain. Although weak modulation is irrelevant, strong modulation induces new ferromagnetic and paramagnetic phases which are fully localized and gapless. The quasiperiodic potential and localized excitations lead to quantum criticality that is intermediate to that of the clean and randomly disordered models with exponents of ν=1^{+} (exact) and z≈1.9, Δ_{σ}≈0.16, and Δ_{γ}≈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 quasiperiodic transition and show its stability to smooth perturbations that preserve the quasiperiodic structure of the model.
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 reference (Crowley et al 2018 Phys. Rev. Lett. 120 175702) 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.
Topological classifications of quantum critical systems have recently attracted growing interest, as they go beyond the traditional paradigms of condensed matter and statistical physics. However, such classifications remain largely unexplored in aperiodic environments, particularly under quasiperiodic modulations. In this Letter, we uncover a family of topological quasiperiodic fixed points with critical scalings intermediate between the clean and infinite-randomness limits. By exactly solving the quasiperiodic cluster-Ising chain, we demonstrate that all phase boundaries separating quasiperiodically modulated phases are governed by topological Ising-like fixed points unique to strongly modulated quasiperiodic systems. Although their local bulk critical properties match those of conventional quasiperiodic Ising critical points reported before, these fixed points host robust topological edge degeneracies and exhibit a distinct symmetry-enriched realization of the quasiperiodic Ising criticality, as further supported by complementary lattice simulations.
Quasiperiodic systems are aperiodic but deterministic, so their critical behavior differs from that of clean systems and disordered ones as well. Quasiperiodic criticality was previously understood only in the special limit where the couplings follow discrete quasiperiodic sequences. Here we consider generic quasiperiodic modulations; we find, remarkably, that for a wide class of spin chains, generic quasiperiodic modulations flow to discrete sequences under a real-space renormalization-group transformation. These discrete sequences are therefore fixed points of a functional renormalization group. This observation allows for an asymptotically exact treatment of the critical points. We use this approach to analyze the quasiperiodic Heisenberg, Ising, and Potts spin chains, as well as a phenomenological model for the quasiperiodic many-body localization transition.
Quantum critical points in quasiperiodic magnets can realize new universality classes, with critical properties distinct from those of clean or disordered systems. Here, we study quantum phase transitions separating ferromagnetic and paramagnetic phases in the quasiperiodic q-state Potts model in 2+1D. Using a controlled real-space renormalization group approach, we find that the critical behavior is largely independent of q, and is controlled by an infinite-quasiperiodicity fixed point. The correlation length exponent is found to be ν=1, saturating a modified version of the Harris-Luck criterion.
We uncover a novel mechanism for inducing a gapful phase in interacting many-body quantum chains. The mechanism is nonperturbative, being triggered only in the presence of both strong interactions and strong aperiodic (disordered) modulation. In the context of the critical antiferromagnetic spin-1/2 XXZ chain, we identify an emerging dimerization, which removes the system from criticality and stabilizes the novel phase. This mechanism is shown to be quite general in strongly interacting quantum chains in the presence of strongly modulated quasiperiodic disorder, which is, surprisingly, perturbatively irrelevant. Finally, we also characterize the associated quantum phase transition via the corresponding critical exponents and thermodynamic properties.
The precise nature of many-body localization (MBL) transitions in both random and quasiperiodic (QP) systems remains elusive so far. In particular, whether MBL transitions in QP and random systems belong to the same universality class or two distinct ones has not been decisively resolved. Here, we investigate MBL transitions in one-dimensional (d=1) QP systems as well as in random systems by state-of-the-art real-space renormalization group (RG) calculation. Our real-space RG shows that MBL transitions in 1D QP systems are characterized by the critical exponent ν≈2.4, which respects the Harris-Luck bound (ν>1/d) for QP systems. Note that ν≈2.4 for QP systems also satisfies the Harris-Chayes-Chayes-Fisher-Spencer bound (ν>2/d) for random systems, which implies that MBL transitions in 1D QP systems are stable against weak quenched disorder since randomness is Harris irrelevant at the transition. We shall briefly discuss experimental means to measure ν of QP-induced MBL transitions.