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Many body Fu-Kane indices

Mahsa Rahnama

Published Sep 4, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.14288/1.0455990

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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.
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.
This thesis studies the mathematical structure of quantum Hall transport in lattice systems, with an emphasis on relating mutiple definitions of Hall conductance in different mathematical branches: Geometrically as a bulk Chern number for a Bloch bundle; analytically as an edge spectral flow for a half-space Hamiltonian; and in operator algebra language as a many body charge transport index generated by magnetic flux insertion. The main results of the thesis reveal restrictions imposed on these indices by frustration-free structure, and improved locality estimates for interacting many body systems.The first result concerns frustration free free-Fermion (F 4 ) Hamiltonians. For translation-invariant, strictly local, gapped F4 Hamiltonians on Z 2 , we prove that the bulk index vanishes. More precisely, the two Bloch bundle associated with the image/kernel of the momentum space Hamiltonian is shown to be trivial by an algebraic argument using polynomial modules and a free resolution. Thus a frustration-free free-Fermion model cannot realize a nontrivial integer quantum Hall phase at the level of the bulk Chern index.The second new result is the corresponding edge statement. For a translation-invariant F4 Hamiltonian restricted to a half-plane, we prove that the edge index vanishes. We first identified that frustration free is equivalent to flat ground state band. Detailed analysis of spectral flow of the edge Hamiltonian shows that the edge mode is an analytic function and can be absorbed by the flat band. Consequently, the edge spectral flow across a regular energy in the bulk gap is constant and must equal its trivial value below the spectrum. This establishes, in this setting, that the frustration-free structure rules out nonzero edge transport as well as nonzero bulk transport. The third new result is an exponential clustering theorem for finite-volume quantum spin systems with a possibly degenerate and split ground-state sector. Assuming only a uniform spectral gap above a low-energy spectral patch of sufficiently small width, we prove exponential decay of correlations in the form|⟨Ω, ABΩ⟩ − ⟨Ω, AP BΩ⟩| ≤ C∥P∥1∥A∥∥B∥e −cd(X,Y ),for local observables A and B with disjoint supports. This improves the locality control needed in many body Hall conductance arguments, where previous estimates under comparable hypotheses were superpolynomial. Together, these results clarify how locality and spectral gaps constrain both free-Fermion and interacting formulations of topological transport.
Geometric bulk-edge correspondence for Z2\mathbb{Z}_2Z2​-topological insulators
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}.
Topological insulators present a bulk gap, but allow for dissipationless spin transport along the edges. These exotic states are characterized by the Z2 topological invariant and are protected by time-reversal symmetry. The Kane–Mele model is one model to realize this topological class in two dimensions, also called the quantum spin Hall state. In this brief review article, we provide a pedagogical introduction to the influence of correlation effects in the quantum spin Hall states, with special focus on the half-filled Kane–Mele–Hubbard model, solved by means of unbiased determinant quantum Monte Carlo (QMC) simulations. We explain the idea of identifying the topological insulator via π-flux insertion, the Z2 invariant and the associated behavior of the zero-frequency Green's function, as well as the spin Chern number in parameter-driven topological phase transitions. The examples considered are two descendants of the Kane–Mele–Hubbard model, the generalized and dimerized Kane–Mele–Hubbard model. From the Z2 index, spin Chern numbers and the Green's function behavior, one can observe that correlation effects induce shifts of the topological phase boundaries. Although the implementation of these topological quantities has been successfully employed in QMC simulations to describe the topological phase transition, we also point out their limitations as well as suggest possible future directions in using numerical methods to characterize topological properties of strongly correlated condensed matter systems.
Here, the authors introduce a set of quantities to diagnose symmetry-protected topological phases of fermions protected by antiunitary symmetries. These quantities, which can be written as expectation values of nonlocal operators, effectively simulate the partition function on nonorientable spacetime manifolds in the operator formalism. The important observation is that the proposed quantities are complex valued for topological states, where the complex phase is the many-body topological invariant. Numerous symmetry classes and dimensionalities are studied in detail. The key ingredient of all quantities is the ``fermionic partial transpose''. For example, the Z2{Z}_{2}Z2​ topological invariant for time-reversal symmetric topological insulators in two spatial dimensions is given.
We show that the two-dimensional Z2{\mathbb{Z}}_{2}Z2​ invariant for time-reversal invariant insulators can be formulated in terms of the boundary-condition dependence of the ground state wave function for both noninteracting and strongly correlated insulators. By introducing a family of quasi-single-particle states associated to the many-body ground state of an insulator, we show that the Z2{\mathbb{Z}}_{2}Z2​ invariant can be expressed as the integral of a certain Berry connection over half the space of boundary conditions, providing an alternative expression to the formulations that appear in S.-S. Lee and S. Ryu [Phys. Rev. Lett. 100, 186807 (2008)]. We show the equivalence of the different many-body formulations of the invariant and show how they reduce to known band-theoretic results for Slater determinant ground states. Finally, we apply our results to analytically calculate the invariant for the Kane-Mele model with nonlocal (orbital) Hatsugai-Kohmoto (HK) interactions. This rigorously establishes the topological nontriviality of the Kane-Mele model with HK interactions and represents one of the few exact calculations of the Z2{\mathbb{Z}}_{2}Z2​ invariant for a strongly interacting system.