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arXiv:1207.0581 · Physical Review B

Persistence of phase boundaries between a topological and trivial Z2 insulator

Zohar Ringel, Ehud Altman

Published Jan 11, 2013• 1 citation
arXivHTMLPDF

Abstract

When time-reversal symmetry is present there is a sharp distinction between topological and trivial band insulators which ensures that, as parameters are varied, these phases are separated by a phase transition at which the bulk gap closes. Surprisingly we find that even in the absence of time-reversal symmetry, gapless regions originating from the phase boundaries persist. Moreover, the critical line generically opens up to enclose Chern insulating phases that are thin but of finite extent in the phase diagram. We explain the topological origin of this effect in terms of quantized charge pumping, showing in particular that it is robust to the effect of disorder and interactions.

DOI
10.1103/physrevb.87.035115

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When time-reversal symmetry is present there is a sharp distinction between topological and trivial band insulators which ensures that, as parameters are varied, these phases are separated by a phase transition at which the bulk gap closes. Surprisingly we find that even in the absence of time-reversal symmetry, gapless regions originating from the phase boundaries persist. Moreover, the critical line generically opens up to enclose Chern insulating phases that are thin but of finite extent in the phase diagram. We explain the topological origin of this effect in terms of quantized charge pumping, showing in particular that it is robust to the effect of disorder and interactions.
We consider two-dimensional Chern insulators and time-reversal invariant topological insulators and discuss the effect of perturbations breaking either particle-number conservation or time-reversal symmetry. The appearance of trivial mass terms is expected to cause quantum phase transitions into trivial phases when such a perturbation overweighs the topological term. These phase transitions are usually associated with a bulk-gap closing. In contrast, the chiral Chern insulator is unaffected by particle-number breaking perturbations. Moreover, the [Formula: see text] topological insulator undergoes phase transitions into topologically trivial phases without bulk-gap closing in the presence of any of such perturbations. In certain cases, these phase transitions can be circumvented and the protection restored by another U(1) symmetry, e.g. due to spin conservation. These findings are discussed in the context of interacting topological insulators.
We study topological insulators under dephasing noise. With examples of both a 2d2d2d Chern insulator and a 3d3d3d topological insulator protected by time-reversal symmetry, we demonstrate that there is a phase transition at finite dephasing strength between phases with nontrivial and trivial topological indices. Here, the topological index is defined through the correlation matrix. The transition can be diagnosed through the spectrum of the whole correlation matrix or of a local subsystem. Interestingly, even if the topological insulator is very close to the topological-trivial critical point in its Hamiltonian, it still takes finite strength of dephasing to change the topological index, suggesting the robustness of topological insulators under dephasing. We further consider Chern insulators in the presence of real-space disorder, which exhibit a ground-state transition between topological and Anderson insulating phases. We find that even strongly disordered Chern insulators, close to the critical disorder strength, exhibit robustness with respect to dephasing.
It is often thought that emergent phenomena in topological phases of matter are destroyed when tuning to a critical point. In particular, topologically protected edge states supposedly delocalize when the bulk correlation length diverges. We show that this is not true in general. Edge states of topological insulators or superconductors remain exponentially localized---despite a vanishing band gap---if the transition increases the topological index. This applies to all classes where the topological classification is larger than Z2Z_2Z2​, notably including Chern insulators. Moreover, these edge states are stable to disorder, unlike in topological semi-metals. This new phenomenon is explained by generalizing band (or mass) inversion---a unifying perspective on topological insulators---to kinetic inversion. In the spirit of the bulk-boundary correspondence, we also identify topological invariants at criticality, which take half-integer values and separate topologically-distinct universality classes by a multi-critical point. This work enlarges the scope of topological protection and stability by showing that bulk energy gaps can be unnecessary. Experimental probes and stability to interactions are discussed.
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