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Journal of Physics Conference Series

Z 2 Topological Anderson Insulator

Ai Yamakage, Kentaro Nomura, Ken‐Ichiro Imura, Yoshio Kuramoto

Published Dec 17, 2012• 2 citations
Publisher

Abstract

Effects of disorder on a two-dimensional Z 2 topological insulator are studied numerically. We propose and study the phase diagram of a variant of Bernevig-Hughes-Zhang model, which takes account of the s z non-conserving spin-orbit coupling. Using scaling analyses, we determine the phase boundary and the critical exponent characterizing the transition between metallic and the topologically insulating phases.

DOI
10.1088/1742-6596/400/4/042070

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Effects of disorder on a two-dimensional Z 2 topological insulator are studied numerically. We propose and study the phase diagram of a variant of Bernevig-Hughes-Zhang model, which takes account of the s z non-conserving spin-orbit coupling. Using scaling analyses, we determine the phase boundary and the critical exponent characterizing the transition between metallic and the topologically insulating phases.
We investigate disorder-driven transitions between trivial and topological insulator (TI) phases in two-dimensional (2D) systems. Our study primarily focuses on the Bernevig-Hughes-Zhang (BHZ) model with Anderson disorder, while other standard 2D TI models exhibit equivalent features. The analysis is based on the local Chern marker (LCM), a local quantity that allows for the characterization of topological transitions in finite and disordered systems. Our simulations indicate that disorder-driven trivial to topological insulator transitions are nicely characterized by C0{\mathcal{C}}_{0}C0​, the disorder-averaged LCM near the central cell of the system. We show that C0{\mathcal{C}}_{0}C0​ is characterized by a single-parameter scaling, namely, ${\mathcal{C}}_{0}(M,W,L)\ensuremath{\equiv}{\mathcal{C}}_{0}(z)$, with $z=[{W}^{\ensuremath{\mu}}\ensuremath{-}{W}_{c}^{\ensuremath{\mu}}(M)]L$, where MMM is the Dirac mass, WWW is the disorder strength, and LLL is the system size, while ${W}_{c}(M)\ensuremath{\propto}\sqrt{M}$ and $\ensuremath{\mu}\ensuremath{\approx}2$ stand for the critical disorder strength and the critical exponent, respectively. Our numerical results are in agreement with a theoretical prediction based on a first-order Born approximation analysis. These observations lead us to speculate that the universal scaling function we have found is rather general for amorphous and disorder-driven topological phase transitions.
Static disorder in a noninteracting gas of electrons confined to two dimensions can drive a continuous quantum (Anderson) transition between a metallic and an insulating state when time-reversal symmetry is preserved but spin-rotation symmetry is broken. The critical exponent $\ensuremath{\nu}$ that characterizes the diverging localization length and the bulk multifractal scaling exponents that characterize the amplitudes of the critical wave functions at the metal-insulator transition do not depend on the topological nature of the insulating state, i.e., whether it is topologically trivial (ordinary insulator) or nontrivial (a Z2{\mathbb{Z}}_{2}Z2​ insulator supporting a quantum spin Hall effect). This is not true of the boundary multifractal scaling exponents, which we show (numerically) to depend on whether the insulating state is topologically trivial or not.
We study the disordered topological Anderson insulator in a two-dimensional (square not strip) geometry. We first report the phase diagram of finite systems and then study the evolution of phase boundaries when the system size is increased to a very large $1120\ifmmode\times\else\texttimes\fi{}1120$ area. We establish that conductance quantization can occur without a bulk band gap, and that there are two distinct scaling regions with quantized conductance: TAI-I with a bulk band gap, and TAI-II with localized bulk states. We show that there is no intervening insulating phase between the bulk conduction phase and the TAI-I and TAI-II scaling regions, and that there is no metallic phase at the transition between the quantized and insulating phases. Centered near the quantized-insulating transition there are very broad peaks in the eigenstate size and fractal dimension d2{d}_{2}d2​; in a large portion of the conductance plateau eigenstates grow when the disorder strength is increased. The fractal dimension at the peak maximum is ${d}_{2}\ensuremath{\approx}1.5$. Effective-medium theory (Coherent Potential Approximation, self-consistent Born approximation) predicts well the boundaries and interior of the gapped TAI-I scaling region, but fails to predict all boundaries save one of the ungapped TAI-II scaling region. We report conductance distributions near several phase transitions and compare them with critical conductance distributions for well-known models.
Topological insulators (superconductors) are materials that host symmetry-protected metallic edge states in an insulating (superconducting) bulk. Although they are well understood, a thermodynamic description of these materials remained elusive, firstly because the edges yield a non-extensive contribution to the thermodynamic potential, and secondly because topological field theories involve non-local order parameters, and cannot be captured by the Ginzburg-Landau formalism. Recently, this challenge has been overcome: by using Hill thermodynamics to describe the Bernevig-Hughes-Zhang model in two dimensions, it was shown that at the topological phase transition the thermodynamic potential does not scale extensively due to boundary effects. Here, we extend this approach to different topological models in various dimensions (the Kitaev chain and Su-Schrieffer-Heeger model in one dimension, the Kane-Mele model in two dimensions and the Bernevig-Hughes-Zhang model in three dimensions) at zero temperature. Surprisingly, all models exhibit the same universal behavior in the order of the topological-phase transition, depending on the dimension. Moreover, we derive the topological phase diagram at finite temperature using this thermodynamic description, and show that it displays a good agreement with the one calculated from the Uhlmann phase. Our work reveals unexpected universalities and opens the path to a thermodynamic description of systems with a non-local order parameter.
The quantum phase transition between the three dimensional Dirac semimetal and the diffusive metal can be induced by increasing disorder. Taking the system of a disordered Z2 topological insulator as an important example, we compute the single particle density of states by the kernel polynomial method. We focus on three regions: the Dirac semimetal at the phase boundary between two topologically distinct phases, the tricritical point of the two topological insulator phases and the diffusive metal, and the diffusive metal lying at strong disorder. The density of states obeys a novel single parameter scaling, collapsing onto two branches of a universal scaling function, which correspond to the Dirac semimetal and the diffusive metal. The diverging length scale critical exponent ν and the dynamical critical exponent z are estimated, and found to differ significantly from those for the conventional Anderson transition. Critical behavior of experimentally observable quantities near and at the tricritical point is also discussed.
Using exact diagonalization and quantum Monte Carlo calculations we investigate the effects of disorder on the phase diagram of both noninteracting and interacting models of two-dimensional topological insulators. In the fermion sign problem-free interacting models we study, electron-electron interactions are described by an on-site repulsive Hubbard interaction and disorder is included via the one-body hopping operators. In both the noninteracting and interacting models we make use of recent advances in highly accurate real-space numerical evaluation of topological invariants to compute phase boundaries and in the noninteracting models determine critical exponents of the transitions. We find different models exhibit distinct stability conditions of the topological phase with respect to interactions and disorder. We provide a general analytical theory that accurately predicts these trends.