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Physical review. B./Physical review. B

Many-body marker for three-dimensional topological insulators with inversion symmetry

Federico Becca, Alberto Parola

Published Jul 24, 2025• 1 citation
Publisher

Abstract

We extend the previously defined many-body marker for two-dimensional Z2 topological insulators [I. Gilardoni et al., Phys. Rev. B 106, L161106 (2022)] to distinguish trivial, weak, and strong topological insulators in three dimensions, in the presence of inversion symmetry. The marker is written in terms of ground-state expectation values of position operators and can be employed to detect topological phases of interacting systems beyond mean-field approximations, e.g., within quantum Monte Carlo techniques. Here we show that the correct results of the noninteracting limit are reproduced by the many-body marker.

DOI
10.1103/yld5-y4cc

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We extend the previously defined many-body marker for two-dimensional Z2 topological insulators [I. Gilardoni et al., Phys. Rev. B 106, L161106 (2022)] to distinguish trivial, weak, and strong topological insulators in three dimensions, in the presence of inversion symmetry. The marker is written in terms of ground-state expectation values of position operators and can be employed to detect topological phases of interacting systems beyond mean-field approximations, e.g., within quantum Monte Carlo techniques. Here we show that the correct results of the noninteracting limit are reproduced by the many-body marker.
We extend the previously defined many-body marker for two-dimensional Z2\mathbb{Z}_2Z2​ topological insulators [I. Gilardoni {\it et al.}, Phys. Rev. B {\bf 106}, L161106 (2022)] to distinguish trivial, weak-, and strong-topological insulators in three dimensions, in presence of the inversion symmetry. The marker is written in term of ground-state expectation values of position operators and can be employed to detect topological phases of interacting systems beyond mean-field approximations, e.g., within quantum Monte Carlo techniques. Here, we show that the correct results of the non-interacting limit are reproduced by the many-body marker.
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 order of Dirac models in any dimension and symmetry class can be mapped to lattice sites by a universal topological marker. From the topological operator that extracts this marker, one can further define a nonlocal marker that detects topological phase transitions. Several prototype models such as the Su-Schrieffer-Heeger model, Majorana chain, Chern insulator, Bernevig-Hughes-Zhang model, chiral and helical ppp-wave superconductors, three-dimensional time-reversal symmetric topological insulators, among many others, are employed to demonstrate the ubiquity of these markers.
Topological states of matter were first introduced for noninteracting fermions on an infinite uniform lattice. Since then, substantial effort has been made to generalize these concepts to more complex settings. Recently, local markers have been developed that can describe the topological state of systems without translational symmetry and a well-defined gap. However, no local marker for interacting matter has been proposed yet that is capable of directly addressing an interacting system. Here, we suggest such a many-body local marker based on the single-particle Green's function. Using this marker we identify topological transitions in finite lattices of a Chern insulator with Anderson disorder and Hubbard interactions. Importantly, our proposal can be generalized to nonequilibrium systems.
We systematically study topological phases of insulators and superconductors (or superfluids) in three spatial dimensions. We find that there exist three-dimensional (3D) topologically nontrivial insulators or superconductors in five out of ten symmetry classes introduced in seminal work by Altland and Zirnbauer within the context of random matrix theory, more than a decade ago. One of these is the recently introduced Z2{\mathbb{Z}}_{2}Z2​ topological insulator in the symplectic (or spin-orbit) symmetry class. We show that there exist precisely four more topological insulators. For these systems, all of which are time-reversal invariant in three dimensions, the space of insulating ground states satisfying certain discrete symmetry properties is partitioned into topological sectors that are separated by quantum phase transitions. Three of the above five topologically nontrivial phases can be realized as time-reversal invariant superconductors. In these the different topological sectors are characterized by an integer winding number defined in momentum space. When such 3D topological insulators are terminated by a two-dimensional surface, they support a number (which may be an arbitrary nonvanishing even number for singlet pairing) of Dirac fermion (Majorana fermion when spin-rotation symmetry is completely broken) surface modes which remain gapless under arbitrary perturbations of the Hamiltonian that preserve the characteristic discrete symmetries, including disorder. In particular, these surface modes completely evade Anderson localization from random impurities. These topological phases can be thought of as three-dimensional analogs of well-known paired topological phases in two spatial dimensions such as the spinless chiral $({p}_{x}\ifmmode\pm\else\textpm\fi{}i{p}_{y})$-wave superconductor (or Moore-Read Pfaffian state). In the corresponding topologically nontrivial (analogous to ``weak pairing'') and topologically trivial (analogous to ``strong pairing'') 3D phases, the wave functions exhibit markedly distinct behavior. When an electromagnetic U(1) gauge field and fluctuations of the gap functions are included in the dynamics, the superconducting phases with nonvanishing winding number possess nontrivial topological ground-state degeneracies.
We employ quantum Monte Carlo techniques to calculate the Z2{Z}_{2}Z2​ topological invariant in a two-dimensional model of interacting electrons that exhibits a quantum spin Hall topological insulator phase. In particular, we consider the parity invariant for inversion-symmetric systems, which can be obtained from the bulk's imaginary-time Green's function after an appropriate continuation to zero frequency. This topological invariant is used here in order to study the trivial-band to topological-insulator transitions in an interacting system with spin-orbit coupling and an explicit bond dimerization. We discuss the accessibility and behavior of this topological invariant within quantum Monte Carlo simulations.