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arXiv:2010.13796 · Physical Review Letters

Delocalization Transition of a Disordered Axion Insulator

Zhida Song, Biao Lian, Raquel Queiroz, Roni Ilan, Bogdan Andrei Bernevig, Ady Stern

Published Jun 29, 2021• 31 citations
arXivHTMLPDF

Abstract

The axion insulator is a higher-order topological insulator protected by inversion symmetry. We show that, under quenched disorder respecting inversion symmetry on average, the topology of the axion insulator stays robust, and an intermediate metallic phase in which states are delocalized is unavoidable at the transition from an axion insulator to a trivial insulator. We derive this conclusion from general arguments, from classical percolation theory, and from the numerical study of a 3D quantum network model simulating a disordered axion insulator through a layer construction. We find the localization length critical exponent near the delocalization transition to be ν=1.42±0.12. We further show that this delocalization transition is stable even to weak breaking of the average inversion symmetry, up to a critical strength. We also quantitatively map our quantum network model to an effective Hamiltonian and we find its low-energy k·p expansion.

DOI
10.1103/physrevlett.127.016602

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The axion insulator is a higher-order topological insulator protected by inversion symmetry. We show that, under quenched disorder respecting inversion symmetry on average, the topology of the axion insulator stays robust, and an intermediate metallic phase in which states are delocalized is unavoidable at the transition from an axion insulator to a trivial insulator. We derive this conclusion from general arguments, from classical percolation theory, and from the numerical study of a 3D quantum network model simulating a disordered axion insulator through a layer construction. We find the localization length critical exponent near the delocalization transition to be ν=1.42±0.12. We further show that this delocalization transition is stable even to weak breaking of the average inversion symmetry, up to a critical strength. We also quantitatively map our quantum network model to an effective Hamiltonian and we find its low-energy k·p expansion.
We construct a three-dimensional (3D), time-reversal symmetric generalization of the Chalker-Coddington network model for the integer quantum Hall transition. In addition to a weak topological insulator phase already accommodated by the network model framework in the preexisting literature, our network model hosts strong topological insulator phases. We unambiguously demonstrate that strong topological insulator phases emerge as intermediate phases between a trivial insulator phase and a weak topological phase. Additionally, we found a duality that relates a trivial insulator phase and a weak topological phase in our network model. Remarkably, strong topological phases are mapped to themselves under this duality. We show that upon adding sufficiently strong disorder, the strong topological insulator phases undergo phase transitions into a metallic phase. We numerically determine the critical exponent of the insulator-metal transition. Our network model explicitly shows how a semiclassical percolation picture of topological phase transitions in two dimensions can be generalized to 3D and opens up a venue for studying 3D topological phase transitions.
Abstract Transitions between distinct obstructed atomic insulators (OAIs) protected by crystalline symmetries, where electrons form molecular orbitals centering away from the atom positions, must go through an intermediate metallic phase. In this work, we find that the intermediate metals will become a scale-invariant critical metal phase (CMP) under certain types of quenched disorder that respect the magnetic crystalline symmetries on average. We explicitly construct models respecting average C2zT, m, and C4zT and show their scale-invariance under chemical potential disorder by the finite-size scaling method. Conventional theories, such as weak anti-localization and topological phase transition, cannot explain the underlying mechanism. A quantitative mapping between lattice and network models shows that the CMP can be understood through a semi-classical percolation problem. Ultimately, we systematically classify all the OAI transitions protected by (magnetic) groups Pm,P2′,P4′Pm,P{2}^{{\prime} },P{4}^{{\prime} }Pm,P2′,P4′ P m , P 2 ′ , P 4 ′ , and P6′P{6}^{{\prime} }P6′ P 6 ′ with and without spin-orbit coupling, most of which can support CMP.
Ensembles that respect symmetries on average exhibit richer topological states than those in pure states with exact symmetries, leading to the concept of average symmetry-protected topological states (ASPTs). The free-fermion counterpart of ASPT is the so-called statistical topological insulator (STI) in disordered ensembles. In this Letter, we demonstrate the existence of an intrinsic STI, which has no clean counterpart. Using a real space construction (topological crystal), we find an axion STI characterized by the average axion angle θ[over ¯]=π, protected by an average C_{4}T symmetry with (C_{4}T)^{4}=1. While the exact C_{4}T symmetry reverses the sign of θ angle, and hence seems to protect a Z_{2} classification of θ=0,π, we prove that the θ=π state cannot be realized in the clean limit if (C_{4}T)^{4}=1. Therefore, the axion STI lacks band insulator correspondence and is thus intrinsic. To illustrate this state, we construct a lattice model and numerically explore its phase diagram, identifying an axion STI phase separated from both band insulators and trivial Anderson insulators by a metallic phase, revealing the intrinsic nature of the STI. We also argue that the intrinsic STI is robust against electron-electron interactions. Our Letter thus provides the first intrinsic crystalline ASPT and its lattice realization.
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.
The effect of a local Kondo coupling and Hubbard interaction on the topological phase of the one-dimensional topological Kondo insulator (TKI) is numerically investigated using the infinite matrix-product state density-matrix renormalization group algorithm. The ground state of the TKI is a symmetry-protected topological (SPT) phase protected by inversion symmetry. It is found that on its own, the Hubbard interaction that tends to force fermions into a one-charge per site order is insufficient to destroy the SPT phase. However, when the local Kondo Hamiltonian term that favors a topologically trivial ground state with a one-charge per site order is introduced, the Hubbard interaction assists in the destruction of the SPT phase. This topological phase transition occurs in the charge sector where the correlation length of the charge excitation diverges while the correlation length of the spin excitation remains finite. The critical exponents, central charge, and the phase diagram separating the SPT phase from the topologically trivial phase are presented.