Symmetric approximant formalism for statistical topological matter

TL;DR

This work addresses the challenge that standard topological invariants fail for disordered systems where protecting symmetries hold only on average. It introduces a symmetric approximant mapping that replaces average symmetries with an exact subgroup symmetry, constructing a locally indistinguishable ensemble on which conventional invariants can be applied. The authors demonstrate mappings for average time-reversal via magnetic translation and average mirror via glide symmetry, and show an intrinsic statistical topological phase detectable by a scattering invariant in a hole-geometry superconductor, highlighting both successes and limitations (e.g., third-order/higher-order hinge states). The approach offers a unified framework to characterize statistical topological matter and guides future work on extending invariants to correlated or interacting disordered systems.

Abstract

The standard approach to characterizing topological matter, computing topological invariants, fails when the symmetry protecting the topological phase is preserved only on average in a disordered system. Because topological invariants rely on enforcing the symmetry exactly, they can overcount phases by incorrectly identifying certain non-robust features as robust. Moreover, in intrinsic statistical topological insulators, enforcing the symmetry exactly is guaranteed to destroy the topological phase. We define a mapping that addresses both issues and provides a unified framework for describing disordered topological matter.

Figures (6)

• Figure 1: An illustration of the symmetric approximant formalism, whose goal is to define a topological invariant $\mathcal{Q}$ for disordered ensembles with average symmetries. Left: A disordered ensemble with an average time-reversal symmetry $\Theta$. Every realization $H_i$ breaks time-reversal symmetry but has an equally likely time-reversed partner $\Theta H_i \Theta^{-1}$ in the ensemble. Right: The corresponding symmetric approximant ensemble where each realization $H_i$ preserves an exact magnetic translation symmetry $\Theta T_{x, L/2}$, and a well-defined topological invariant $\mathcal{Q}'(H_i)$. The arrow between the ensembles indicates the mapping from the original ensemble to the symmetric approximant ensemble.
• Figure 2: Disordered supercells with various symmetries at length scales larger than the localization length. (a) Disordered sample without any symmetry. (b) The disorder realization in (a) tiled with supercell translation symmetry. (c) A disordered supercell with magnetic translation symmetry.
• Figure 3: Invariant and conductances in mirror- and glide-symmetric systems. (a) Disorder sample with mirror symmetry. (b) The median of conductance with the 16th-84th percentile band of a square with mirror- and glide-symmetric disorder and of a square with disorder that does not respect any symmetry. (c) Disorder sample with glide symmetry. (d) The mean of the topological invariant with $\pm \sigma$ for a system with mirror and glide symmetry. For the computation in (b/d) we use $\delta=0.1$, $\alpha = 0.5$, $L_{x,y}=200$ and 50 seeds.
• Figure 4: Invariant and conductance in a rotationally symmetric square with a hole. (a) Median determinants with 16th-84th percentile bands of the $\mathcal{I}$-symmetric blocks of the reflection matrix $r$ for a square system with a hole. The insets show a diagram of a Corbino disk with and without topological edge states. (b) The median conductance with the 16th-84th percentile bands of the same system. For this computation, we use $\delta=0.1$, $M = 0.7$, $L_{x,y}=300$, $R_{hole}=1$, and 50 seeds. The inset shows the determinant of one of the $\mathcal{I}$-symmetric blocks as a function of both $\alpha$ and $M$ in a system with $L_{x,y}=200$, $R_{hole}=1$, $\delta=0.1$ and for a single seed.
• Figure 5: Possible relations between the topological classifications of the ensemble with an exact symmetry (purple region with a border), ensemble with an average symmetry (blurry purple region), and the symmetry class of the approximant (blue region that encloses the purple region). The orange lines show delocalization transitions. (a) The phase transition of the approximant symmetry class probes the phase transition of the disordered ensemble (Sec. \ref{['sec:magnetictranslation']}). (b) The approximant ensemble admits a continuous deformation between topologically distinct phases of the original ensemble (dashed path). (c) The classification of the ensemble with the average symmetry is reduced compared to the exact one, and is captured by the approximant ensemble (Sec. \ref{['sec:indistinguishability']}). (d) Some of the phase transitions of the symmetry class of the approximant do not correspond to those of the original ensemble.