Teo and Kane (Phys. Rev. B 82, 115120, 2010) classified topological defects (vortices, dislocations, domain walls) in band insulators and superconductors by treating the Bloch/Bogoliubov-de Gennes Hamiltonian as an adiabatically varying function H(k,r)H(\mathbf{k},\mathbf{r}) of crystal momentum k\mathbf{k} around the defect, and proved a bulk-boundary correspondence relating topological invariants of H(k,r)H(\mathbf{k},\mathbf{r}) to protected defect-bound gapless modes; the relation takes the form of an index theorem only in the chiral symmetry classes. Their stated open questions are to establish this correspondence without the framework's two load-bearing restrictions: (i) prove a general index theorem connecting the topological invariants to an analytic index counting defect-localized zero modes for the classes without chiral symmetry that carry Z2\mathbb{Z}_2 invariants, valid beyond the adiabatic approximation (including disorder); (ii) formulate practical defect invariants applicable to interacting and strongly disordered systems, where single-particle bands are absent; and (iii) extend the defect classification to inherently correlated, fractionalized phases, including invariants for fractionalized defect-bound states such as fractional-quantum-Hall-like edge modes on line defects. Later work resolved well-defined special cases (a mod-2 index for class D point defects; Green's-function invariants for symmetry-free 3D line defects beyond the semiclassical limit; defect theories for Abelian topological order and for 'genons' — twist defects binding fractionalized edge-like modes — in multilayer fractional quantum Hall systems; many-body invariants for bulk symmetry-protected phases), but a general index theorem for the non-chiral classes and a unified practical invariant formulation covering interacting, strongly disordered, and fractionalized systems remain open.

Background

A topological insulator or topological superconductor is a gapped phase whose gapless boundary excitations are protected by symmetries rather than by local order parameters. In the single-particle description, such phases are classified within the 'tenfold way' of Altland and Zirnbauer, which sorts single-particle Hamiltonians — Bloch Hamiltonians for insulators, or Bogoliubov-de Gennes (BdG) Hamiltonians for superconductors, the latter encoding the superconducting quasiparticle problem — by the presence or absence of time-reversal symmetry (TRS), particle-hole symmetry (PHS, the BdG redundancy relating quasiparticle partners), and chiral symmetry (the product of TRS and PHS, which anticommutes with the Hamiltonian). Schnyder, Ryu, Furusaki and Ludwig, and Kitaev showed that these symmetry classes support topological invariants (integer winding or Chern numbers in the chiral classes, Z2\mathbb{Z}_2 parities in classes such as D and DIII) that cannot change without closing the bulk gap, and a bulk-boundary correspondence states that a boundary between regions with different invariants hosts a compensating number of protected gapless modes.

Teo and Kane (2010) extended this correspondence from flat boundaries to topological defects inside a 3D crystal: vortices in superconductors, lattice dislocations, and domain walls. Their framework models a defect by a Bloch/BdG Hamiltonian H(k,r)H(\mathbf{k},\mathbf{r}) that depends on crystal momentum k\mathbf{k} and on adiabatic spatial parameters r\mathbf{r} labeling loops or spheres around the defect, with the assumption that HH varies slowly on the scale of the lattice. The homotopy classes of such maps are the defect's topological classes, and explicit formulas for the invariants were given for all ten symmetry classes. The framework predicts, for example, chiral and helical Majorana modes on superconducting vortex lines and Majorana zero modes (zero-dimensional, particle-hole self-conjugate bound states relevant to Ising-type non-Abelian statistics) on defect endpoints. For the chiral classes the correspondence is a physical avatar of a mathematical index theorem, of the Jackiw-Rossi/Callias type (Dirac-operator index theorems equating the net number of localized zero modes to a topological winding of the mass texture): the analytic index, the net number of defect-localized zero modes, equals a topological index computed from the invariant. For the classes without chiral symmetry, whose invariants are Z2\mathbb{Z}_2 parities (for example class D, the symmetry class of the Kitaev chain — the exactly solvable one-dimensional p-wave superconducting wire model — and of a p-wave superconductor vortex), no such analytic index was available in Teo and Kane's adiabatic treatment.

The construction has two structural restrictions. First, it is adiabatic and band-based: it needs a slowly varying, clean single-particle Hamiltonian, so strong disorder or genuine spatial inhomogeneity lies outside the controlled derivation (noncommutative-geometry reformulations exist for disordered but noninteracting bulk/edge problems, as in Prodan and Schulz-Baldes' book). Second, it is noninteracting: Fidkowski and Kitaev (2010) famously showed that interactions collapse the free-fermion classification of the one-dimensional class BDI chain from Z\mathbb{Z} to Z8\mathbb{Z}_8, so invariants built from Bloch/BdG bands cannot be the whole story once interactions are present. Finally, the framework cannot describe inherently correlated phases such as the Laughlin fractional quantum Hall (FQH) state, whose excitations are anyons carrying fractional charge and statistics rather than weakly interacting band fermions; whether FQH-type edge modes can arise as defect-bound modes, and what invariant would characterize them, was posed as an open question. The problem below asks for the general, interaction- and disorder-robust mathematical formulation of this defect bulk-boundary correspondence.

Problem Statement

Establish the defect bulk-boundary correspondence of Teo and Kane (2010) in a general, interaction- and disorder-robust form, removing the adiabatic single-particle band-structure restriction of the original framework. Specifically, within the tenfold-way symmetry classes of Bloch/Bogoliubov-de Gennes Hamiltonians, and then beyond that framework:

(1) Prove a general index theorem that relates the topological invariants assigned to a defect Hamiltonian H(k,r)H(\mathbf{k},\mathbf{r}) to an analytic index counting the defect-localized zero modes, for the symmetry classes without chiral symmetry whose invariants are Z2\mathbb{Z}_2 (parities), and valid beyond the adiabatic approximation of slowly spatially varying bands, including genuinely inhomogeneous or disordered systems.

(2) Formulate practical topological invariants that classify defects and their protected gapless modes in interacting and/or strongly disordered systems, in a manner that does not rely on single-particle Bloch/BdG band structure or the adiabatic spatial-variation assumption, and that reproduces and extends the free-fermion defect classification (accounting, for example, for interaction-driven reductions of the free classification such as the Fidkowski-Kitaev collapse).

(3) Generalize the defect classification to inherently correlated topological textures: determine whether fractionalized excitations (for instance, fractional quantum Hall edge states or other fractionalized modes) can arise as topological line defects in higher-dimensional systems, and construct the topological invariants that characterize such fractionalized defect-bound states.

These three components jointly discharge one scientific objective: putting the protected defect-bound gapless modes of topological insulators and superconductors on the same rigorous footing as the ordinary bulk-boundary correspondence, without assuming adiabaticity, absence of disorder, absence of interactions, or band quasiparticles. The heterostructure-realization direction raised in the same source conclusion is a downstream materials application of this objective rather than one of the three components above.

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