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arXiv:2606.27318

Geometric bulk-edge correspondence for $\mathbb{Z}_2$-topological insulators

Alexis Drouot, Jacob Shapiro, Xiaowen Zhu

math-phmath.SPquant-phSubmitted Jun 25, 2026• Announced Jun 26, 2026
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Abstract

Fermionic time-reversal-invariant insulators in two dimensions--class AII in the Kitaev table--come in two topological phases. These phases are characterized by a $\mathbb{Z}_2$-valued invariant, the Fu-Kane-Mele index. We prove a geometric bulk-edge correspondence for curved interfaces: if two such insulators occupy complementary regions separated by a curved boundary, then the $\mathbb{Z}_2$ edge index of the interface system is the product, modulo two, of the difference of the two bulk $\mathbb{Z}_2$ indices and a geometric intersection number associated with the boundary and the measurement region. The argument is a $\mathbb{Z}_2$ analogue of the curved-interface connection formula proved for Hall insulators in \cite{DZ24}.

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21 pages, 5 figures

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Fermionic time-reversal-invariant insulators in two dimensions--class AII in the Kitaev table--come in two topological phases. These phases are characterized by a Z2\mathbb{Z}_2Z2​-valued invariant, the Fu-Kane-Mele index. We prove a geometric bulk-edge correspondence for curved interfaces: if two such insulators occupy complementary regions separated by a curved boundary, then the Z2\mathbb{Z}_2Z2​ edge index of the interface system is the product, modulo two, of the difference of the two bulk Z2\mathbb{Z}_2Z2​ indices and a geometric intersection number associated with the boundary and the measurement region. The argument is a Z2\mathbb{Z}_2Z2​ analogue of the curved-interface connection formula proved for Hall insulators in \cite{DZ24}.
The topological classification of fermion systems in mixed states is a long-standing quest. For Gaussian states, reminiscent of noninteracting unitary fermions, some progress has been made. While the topological quantization of certain observables such as the Hall conductivity is lost for mixed states, directly observable many-body correlators exist which preserve the quantized nature and naturally connect to known topological invariants in the ground state. For systems that break time-reversal (TR) symmetry, the ensemble geometric phase was identified as such an observable which can be used to define a Chern number in (1+1)(1+1)(1+1) and two dimensions. Here we propose a corresponding Z2{\mathbb{Z}}_{2}Z2​ topological invariant for systems with TR symmetry. We show that this mixed-state invariant is identical to well-known Z2{\mathbb{Z}}_{2}Z2​ invariants for the ground state of the so-called fictitious Hamiltonian, which for thermal states is just the ground state of the system Hamiltonian itself. We illustrate our findings for finite-temperature states of a paradigmatic Z2{\mathbb{Z}}_{2}Z2​ topological insulator, the Kane-Mele model.
Chiral-antichiral pairs of non-Abelian Hall states, like the Pfaffian, Read-Rezayi and NASS states, can be used to model two-dimensional time-reversal invariant topological insulators. Their stability was shown to be associated to the presence of a Z2 anomaly and characterized by the same Z2 index introduced for free fermion and Abelian systems. In this work, we continue the stability analysis by providing the form of time-reversal invariant interactions that gap the non-Abelian edge excitations. Our approach is based on the description of non-Abelian states as projections of corresponding “parent ” Abelian states. ar X iv
Two-dimensional topological insulators possess conducting edge states at their boundary while being insulating in the bulk. We investigate the edge state emergent at a smooth topological phase boundary of interacting fermions within a full real-space analysis of the time-reversal invariant Hofstadter-Hubbard model. We characterize the localization of the edge state and the topological phase boundary by means of the local compressibility, the spectral density, a generalized local spin Chern marker as well as the Hall response and find good agreement between all these quantities. Computing the edge state spectra at the interface we observe robustness of the edge state against fermionic two-body interactions and conclude that interactions only shift its position. Hence the bulk-boundary correspondence for the interacting system is confirmed. Since experimental probing of edge states remains a challenge in ultracold atom setups, we propose the detection of the local compressibility by measuring correlations with a quantum gas microscope.
Topological insulators are noninteracting, gapped fermionic systems which have gapless boundary excitations. They are characterized by topological invariants, which can be written in many different ways, including in terms of Green's functions. Here we show that the existence of the edge states directly follows from the existence of the topological invariant written in terms of the Green's functions, for all ten classes of topological insulators in all spatial dimensions. We also show that the resulting edge states are characterized by their own topological invariant, whose value is equal to the topological invariant of the bulk insulator. This can be used to test whether a given model Hamiltonian can describe an edge of a topological insulator. Finally, we observe that the results discussed here apply equally well to interacting topological insulators, with certain modifications.
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In this thesis, we construct a many-body Z₂ index for two-dimensional interacting fermionic lattice systems with a conserved U(1) charge and fermionic time-reversal symmetry. The construction is formulated within the operator-algebraic framework of quantum lattice systems and applies to symmetric short-range entangled states. Motivated by the flux-insertion interpretation of the Fu–Kane–Mele invariant, we define defect states by adiabatically inserting a π-flux through the system and analyze their transformation properties under time reversal. The index takes its non-trivial value precisely when the resulting flux defect transforms as part of a Kramers pair. We prove that the index is independent of all auxiliary choices involved in its definition, including the choice of parent Hamiltonian and flux-insertion geometry. Furthermore, we show that it is multiplicative under stacking and invariant under symmetry-preserving locally generated automorphisms. Consequently, the index depends only on the underlying symmetry-protected topological phase and extends naturally to the class of symmetric stably short-range entangled states. To connect the many-body construction with the established free-fermion theory, we develop a CAR-algebraic formulation of the Fu–Kane–Mele invariant based on flux insertion and spectral flow. We prove that, for symmetric quasi-free states in symmetry class AII, the many-body index coincides with the Fu–Kane–Mele invariant. As a consequence, the interacting index recovers the non-interacting classification and establishes the persistence of the quantum spin Hall phase in the presence of interactions. These results provide a rigorous operator-algebraic characterization of interacting two-dimensional fermionic symmetry-protected topological phases with time-reversal symmetry and furnish a many-body extension of the Fu–Kane–Mele invariant.