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arXiv:2409.12120 · Physical review. B./Physical review. B

Computing the Z 2 invariant in two-dimensional strongly correlated systems

Sounak Sinha, Derek Y. Pan, Barry Bradlyn

Published Feb 6, 2025• 7 citations
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Abstract

We show that the two-dimensional ${\mathbb{Z}}_{2}$ 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 ${\mathbb{Z}}_{2}$ 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 ${\mathbb{Z}}_{2}$ invariant for a strongly interacting system.

DOI
10.1103/physrevb.111.085116

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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.
Taking the clue from the modern theory of polarization [Rev. Mod. Phys. 66, 899 (1994)], we identify an operator to distinguish between Z2{\mathbb{Z}}_{2}Z2​-even (trivial) and Z2{\mathbb{Z}}_{2}Z2​-odd (topological) insulators in two spatial dimensions. Its definition extends the position operator [Phys. Rev. Lett. 82, 370 (1999)], which was introduced in one-dimensional systems. We first show a few examples of noninteracting models where single-particle wave functions are defined and allow for a direct comparison with standard techniques on large system sizes. Then, we illustrate its applicability for an interacting model on a small cluster where exact diagonalizations are available. Its formulation in the Fock space allows a direct computation of expectation values over the ground-state wave function (or any approximation of it), thus, allowing us to investigate generic interacting systems, such as strongly correlated topological insulators.
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.
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.
Inspired by a recently constructed commuting-projector Hamiltonian for a two-dimensional (2D) time-reversal-invariant topological superconductor [Z. Wang et al., Phys. Rev. B 98, 094502 (2018)], we introduce a commuting-projector model that describes an interacting yet exactly solvable 2D topological insulator. We explicitly show that both the gapped and gapless boundaries of our model are consistent with those of band-theoretic, weakly interacting topological insulators. Interestingly, on certain lattices our time-reversal-symmetric models also enjoy CP\mathcal{CP}CP symmetry, leading to intuitive interpretations of the bulk invariant for a CP\mathcal{CP}CP-symmetric topological insulator upon putting the system on a Klein bottle. We also briefly discuss how these many-body invariants may be able to characterize models with only time-reversal symmetry.
For interacting Z2{Z}_{2}Z2​ topological insulators with inversion symmetry, we propose a simple topological invariant expressed in terms of the parity eigenvalues of the interacting Green's function at time-reversal invariant momenta and zero frequency. We derive this result from our previous formula involving the integral over the frequency-momenta space. This formula greatly simplifies the explicit calculation of Z2{Z}_{2}Z2​ topological invariants in inversion-symmetric insulators with strong interactions.
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.