Topological Anderson insulators by latent symmetry

Abstract

Topological Anderson insulators represent a class of disorder-induced, nontrivial topological states of matter. In this study, we propose a feasible strategy to unveil and design topological Anderson insulators protected by latent symmetries. These are not visible in the original system, but become obvious after performing an isospectral reduction. Using this technique, we design a family of disordered multi-atomic chains that exhibit latent chiral symmetry or mirror (inversion) symmetry. Using topological invariants, bulk polarization, and the divergence of localization length of the topological bound edge states in the reduced disordered system, we show how to identify the gapped and ungapped topological Anderson states in the original systems. Our work thus extends the concept of topological Anderson insulating phases protected by geometric symmetries and tenfold-way classification to the various types of latent symmetry cases. Overall, our work paves the way for exploiting topological Anderson insulators in terms of latent symmetries.

Figures (5)

• Figure 1: A series of lattice building blocks $G_0$, $G_1$, $G_2$, and $G_3$ contains lattice sites $S = \{A,B\}$ that are either mirror symmetric (a) or latently mirror symmetric (b). (c) Two building blocks and their isospectral reduction. The clouds represent an arbitrary system. (d) Gluing together the building blocks from (c) at site $B$ results in a system whose isospectral reduction is directly related to that of the isolated building blocks (see text for details). (e) A one-dimensional chain formed by (possibly different) building blocks, glued together at sites $A$ or $B$. (f) Performing the isospectral reduction over sites $S=\{A,B\}$ yields a one-dimensional chain (not necessarily periodic) of sites $A,B$, with energy-dependent on-site potentials $V_{A,j}$, and $V_{B,j}$, and energy-dependent couplings $a_j(E)$ and $b_j(E)$; see text for details.
• Figure 2: (a) Trimer chain, with sites in the unit cell marked as $A$, $B$, and $C$. The intercell hopping amplitude is $t$, and the intracell hopping amplitudes are $J_j$ and $t$. (b) Two building blocks and isospectral reduction of $G_1$. (c) Gluing together the building blocks from (b) at site $B$ results in a system. (d) Phase diagram, showing the disorder-averaged real-space topological number $\mathcal{Q}$ in the $W -J$ plane for the disordered trimerized model at filling factor $1/3$ with $t = 1$, and $N=1000$. The green (yellow) region stands for the topological Anderson gapped (ungapped) insulator phase. The blue region indicates the trivial insulator phase. (e) The divergence of the localization length $\Lambda$ of topological edge states, for the same system parameters as in (d). The white line represents the topological phase boundary. (f) Disorder-averaged topological number $\mathcal{Q}$ and (g) eigenvalues $E$ as a function of $W$ at the initial hopping amplitude $J=1.05$ of (d), where the number of unit cells is fixed as $N=2000$. Red lines highlight the $N$th eigenvalue, and blue lines show all others. In both (f) and (g), white, green, and yellow shades denote the trivial phase, the gapped TAI phase with bulk energy gap, and the ungapped TAI phase without bulk energy gap, respectively.
• Figure 3: (a) Tetra-atomic chain consisting of a four site unit cell, with intracell hopping amplitudes $J$ (black thin solid line), $J_a$ (green solid line), and $J_b$ (blue solid line). (b,c) Phase diagrams at fillings $1/4$, and $3/4$, respectively, of a tetra-atomic chain with correlated random disorder for intercell hopping $t = 1$.
• Figure 4: (a) Schematic of a tetra-atomic chain with random flux. (b,c) Phase diagrams at fillings $1/4$, and $3/4$, respectively, of a tetra-atomic chain with random flux and with intercell hopping $t = 1$.
• Figure 5: (a) Octatomic chain composed of eight lattice points per unit cell. The intercell hopping amplitude is $t$, and the intracell hopping amplitudes are $J$, $J_{a,j}$, and $J_{b,j}$. (b) Topological invariant $P_0$ and (c) bulk energy gap $\Delta E$ in the $(W, J)$ plane at filling $1/8$ for $t = 1$, $N = 200$, and $N_{c} = 100$. The red (blue) solid line indicates the phase boundary determined by $\Delta E=3 \times 10^{-2}$. (d) Energy gap $\Delta E$ and bulk polarization $P_0$ as a function of $J$ for the case when $N = 200$, $N_c=100$, $t=1$, and $W=0.65$.