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arXiv:1909.05857 · Physical review. B./Physical review. B

Anomalous localization at the boundary of an interacting topological insulator

Itamar Kimchi, Yang-Zhi Chou, Rahul Nandkishore, Leo Radzihovsky

Published Jan 17, 2020• 12 citations
arXivHTMLPDF

Abstract

The boundary of a topological insulator (TI) hosts an anomaly restricting its possible phases: e.g., three-dimensional (3D) strong and weak TIs maintain surface conductivity at any disorder if symmetry is preserved on average at least when electron interactions on the surface are weak. However, the interplay of strong interactions and disorder with the boundary anomaly has not yet been theoretically addressed. Here we study this combination for the edge of a two-dimensional TI and the surface of a 3D weak TI, showing how it can lead to an ``Anomalous Many Body Localized'' (AMBL) phase that preserves the anomaly. We discuss how the anomalous Kramers parity switching with $\ensuremath{\pi}$ flux arises in the bosonized theory of the localized helical state. The anomaly can be probed in localized boundaries by electrostatically sensing nonlinear hopping transport with $e/2$ shot noise. Our AMBL construction in 3D weak TIs fails for 3D strong TIs, which suggests that their anomaly restrictions are distinguished by strong interactions.

DOI
10.1103/physrevb.101.035131

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The boundary of a topological insulator (TI) hosts an anomaly restricting its possible phases: e.g., three-dimensional (3D) strong and weak TIs maintain surface conductivity at any disorder if symmetry is preserved on average at least when electron interactions on the surface are weak. However, the interplay of strong interactions and disorder with the boundary anomaly has not yet been theoretically addressed. Here we study this combination for the edge of a two-dimensional TI and the surface of a 3D weak TI, showing how it can lead to an ``Anomalous Many Body Localized'' (AMBL) phase that preserves the anomaly. We discuss how the anomalous Kramers parity switching with \ensuremathπ\ensuremath{\pi}\ensuremathπ flux arises in the bosonized theory of the localized helical state. The anomaly can be probed in localized boundaries by electrostatically sensing nonlinear hopping transport with e/2e/2e/2 shot noise. Our AMBL construction in 3D weak TIs fails for 3D strong TIs, which suggests that their anomaly restrictions are distinguished by strong interactions.
We study the edge and surface theories of topological insulators from the perspective of anomalies and identify a Z2{\mathbb{Z}}_{2}Z2​ anomaly associated with charge conservation. The anomaly is manifested through a two-point correlation function involving creation and annihilation operators on two decoupled boundaries. Although charge conservation on each boundary requires this quantity to vanish, we find that it diverges. A corollary result is that under an insertion of a flux quantum, the ground state evolves to an exactly orthogonal state independent of the rate at which the flux is inserted. The anomaly persists in the presence of disorder and imposes sharp restrictions on possible low-energy theories. Being formulated in a many-body, field-theoretical language, the anomaly allows one to test the robustness of topological insulators to interactions in a concise way.
The surfaces of three-dimensional topological insulators (3D TIs) are generally described as Dirac metals, with a single Dirac cone. It was previously believed that a gapped surface implied breaking of either time-reversal T\mathcal{T}T or U(1)\text{U}(1)U(1) charge conservation symmetry. Here, we discuss a possibility in the presence of interactions, a surface phase that preserves all symmetries but is nevertheless gapped and insulating. Then, the surface must develop topological order of a kind that can not be realized in a two-dimensional (2D) system with the same symmetries. We discuss candidate surface states, non-Abelian quantum Hall states which, when realized in 2D, have ${\ensuremath{\sigma}}_{xy}=\frac{1}{2}$ and hence break T\mathcal{T}T symmetry. However, by constructing an exactly soluble 3D lattice model, we show they can be realized as T\mathcal{T}T-symmetric surface states. The corresponding 3D phases are confined, and have $\ensuremath{\theta}=\ensuremath{\pi}$ magnetoelectric response. Two candidate states have the same 12-particle topological order, the (Read-Moore) Pfaffian state with the neutral sector reversed, which we term T-Pfaffian topological order, but differ in their T\mathcal{T}T transformation. Although we are unable to connect either of these states directly to the superconducting TI surface, we argue that one of them describes the 3D TI surface, while the other differs from it by a bosonic topological phase. We also discuss the 24-particle Pfaffian-antisemion topological order (which can be connected to the superconducting TI surface) and demonstrate that it can be realized as a T\mathcal{T}T-symmetric surface state.
Topological insulators (TIs) are an exciting discovery because of their robustness against disorder and interactions. Recently, second-order TIs have been attracting increasing attention, because they host topologically protected 1D hinge states in 3D or 0D corner states in 2D. A significantly critical issue is whether the second-order TIs also survive interactions, but it is still unexplored. We study the effects of weak Coulomb interactions on a 3D second-order TI, with the help of renormalization-group calculations. We find that the 3D second-order TIs are always unstable, suffering from two types of topological phase transitions. One is from second-order TI to TI, the other is to normal insulator. The first type is accompanied by emergent time-reversal and inversion symmetries and has a dynamical critical exponent κ=1. The second type does not have the emergent symmetries but has nonuniversal dynamical critical exponents κ<1. Our results may inspire more inspections on the stability of higher-order topological states of matter and related novel quantum criticalities.
Two-dimensional topological insulators, characterized by symmetry-protected anomalous boundary modes, have been generalized to the strongly correlated regime for both bosonic and fermionic systems. As correlated topological insulators (TI) approach experimental realization in quantum simulators, conventional probes, such as transport measurements, are not easily applicable to these synthetic platforms. In this study, we focus on two examples of correlated TI: a bosonic TI protected by Z2×U(1)\mathbb{Z}_2\times U(1)Z2​×U(1) symmetry and the fermionic quantum spin Hall insulator protected by time-reversal symmetry. We propose a unified, readily implementable protocol based on measuring the disorder parameter ⟨exp⁡(iθQ^A)⟩\langle\exp(\mathrm{i}θ\hat{Q}_A)\rangle⟨exp(iθQ^​A​)⟩ for a large subregion AAA, with Q^A\hat{Q}_AQ^​A​ the total charge operator within AAA. Our key finding is that this quantity exhibits non-analytical dependence on θθθ for correlated TI, a signature robust against decoherence. We establish this diagnostic through both numerical simulations and analytical derivations. This protocol is well-suited for implementation on near-term quantum simulation platforms, providing a direct route to experimentally confirm correlated TI.
We show that boundaries of 3D weak topological insulators can become gapped by strong interactions while preserving all symmetries, leading to Abelian surface topological order. The anomalous nature of weak topological insulator surfaces manifests itself in a nontrivial action of symmetries on the quasiparticles; most strikingly, translations change the anyon types in a manner impossible in strictly 2D systems with the same symmetry. As a further consequence, screw dislocations form non-Abelian defects that trap Z_{4} parafermion zero modes.
We show that the gapless boundary signatures---namely, chiral/helical hinge modes or localized zero modes---of three-dimensional higher-order topological insulators and superconductors with inversion symmetry can be gapped without symmetry breaking upon the introduction of non-Abelian surface topological order. In each case, the fractionalization pattern that appears on the surface is ``anomalous'' in the sense that it can be made consistent with symmetry only on the surface of a three-dimensional higher-order insulator/superconductor. Our results show that the interacting manifestation of higher-order topology is the appearance of ``anomalous gapped boundaries'' between distinct topological orders whose quasiparticles are related by inversion, possibly in conjunction with other protecting symmetries such as time-reversal symmetry and charge conservation.