Topology of ultra-localized insulators and superconductors

TL;DR

This work introduces a framework to classify ultra-localized insulators (TUIs) — fully localized bulk systems — by mapping their localized eigenbasis to an auxiliary chiral Hamiltonian and applying the tenfold-way classification. It demonstrates a threefold TUI structure tied to the chiral classes $AIII$, $BDI$, and $CII$, with a special treatment for $d=1$ and Bott periodicity for $d\ge2$, and shows TUIs can host robust, delocalized boundary states despite a fully localized bulk. The authors connect TUI topology to conventional TI topology via a group homomorphism, clarifying which conventional topological phases can be Wannierized and identifying new fully localized topological phases absent in the standard classification. A concrete 3D TUI example in class $AII$ is constructed, highlighting anomalous surface states that persist under strong boundary disorder and illustrating the practical implications for Wannierizability and the broader topology of disordered quantum matter.

Abstract

The topology of an insulator can be defined even when all eigenstates of the system are localized - an extreme case of Anderson insulators that we call ultra-localized. We derive the classification of such ultra-localized insulators in all symmetry classes and dimensions. We clarify their bulk-boundary correspondence and show that ultra-localized systems are in many instances phases of matter not described by the known classification of topological insulators and superconductors. As a consequence, we clarify which conventional topological phases are Wannierizable, and which topological phases cannot exist without delocalized states.

Figures (4)

• Figure 1: Localization properties of eigenstates of generic disordered systems across topological phase transitions. (a) Phase transition between a topological (ultra localized) insulator and a trivial insulator in $d=1$, driven by a parameter $\lambda$. In both phases all states are localized (grey), and critical states (blue) appear at zero energy Essin_2015bagrets2016 at the transition point. (b) Phase transition out of a disordered TI (hatched region) in $d=2$, not deformable into an ultra-insulator. The states in red are metallic. A trivial ultra-insulator emerges at strong disorder as the delocalized states pair-annihilate, while TIs appear only at specific fillings and require bulk delocalized states. Here, $W$ is the disorder strength. (c) Transition between a topological and a trivial ultra-insulator in $d=3$. There is an extended critical region characterized by delocalized states appearing somewhere in the bulk spectrum.
• Figure 2: Sketch of the surface of a time-reversal invariant TUI in $d=3$. While its bulk consists only of fully localized orbitals (grey states), its anomalous surface can be understood as a single band of a $\mathbb{Z}_2$ TI in $d=2$. In particular, if the energy windows (red) supporting delocalized states for two adjacent surfaces I and II do not overlap, delocalized helical hinge states between the two surfaces are found at any energy $E_{\mathrm{F}}$ between the red windows of delocalized surface states.
• Figure 3: A TUI in a thick Corbino disk geometry, threaded by a magnetic flux $\Phi$ (a) and its spectrum as a function of $\Phi$ (b). Hinge modes are indicated in red. To bring about the hinge modes, energies of the states localized near the vertical ($\perp x$) and horizontal ($\perp y$) surfaces are shifted by $E_x$ and $E_y$, respectively. Because there are four hinges, each dispersing branch in (b) is fourfold degenerate.
• Figure 4: Conventional topological phases come in two flavors: (a) Non-localizable TIs, such as the quantum Hall or quantum spin Hall insulator, necessarily have extended bulk states below and above the Fermi level. Disappearance of these extended states is always accompanied by a transition to a different (usually trivial) phase. (b) Wannier-localizable TIs may have extended states above or below the Fermi level, but these states can be localized by varying some parameter $\lambda$ without a phase transition, i.e., the band of extended states does not enclose the topological phase.