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arXiv:1303.3425 · Physical Review B

Z2 topological invariants in two dimensions from quantum Monte Carlo

Thomas C. Lang, Andrew M. Essin, Victor Gurarie, Stefan Weßel

Published May 1, 2013• 48 citations
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Abstract

We employ quantum Monte Carlo techniques to calculate the ${Z}_{2}$ 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.

DOI
10.1103/physrevb.87.205101

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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.
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.
The aim of this series of two papers is to discuss topological invariants for interacting topological insulators (TIs). In the first paper (I), we provide a paradigm of efficient numerical evaluation scheme for topological invariants, in which we demystify the procedures and techniques employed in calculating Z2{Z}_{2}Z2​ invariant and spin Chern number via zero-frequency single-particle Green's function in quantum Monte Carlo (QMC) simulations. Here we introduce an interpolation process to overcome the ubiquitous finite-size effect, so that the calculated spin Chern number shows ideally quantized values. We also show that making use of symmetry properties of the underlying systems can greatly reduce the computational effort. To demonstrate the effectiveness of our numerical evaluation scheme, especially the interpolation process, for calculating topological invariants, we apply it on two independent two-dimensional models of interacting topological insulators. In the subsequent paper (II), we apply the scheme developed here to wider classes of models of interacting topological insulators, for which certain limitation of constructing topological invariant via single-particle Green's functions will be presented.
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.
Topological phase transitions in free fermion systems can be characterized by the closing of single-particle gap and the change in topological invariants. However, in the presence of electronic interactions, topological phase transitions can be more complicated. In paper I of this series [Phys. Rev. B 93, 195163 (2016)], we have proposed an efficient scheme to evaluate the topological invariants based on the single-particle Green's function formalism. Here, in paper II, we demonstrate several interaction-driven topological phase transitions (TPTs) in two-dimensional (2D) interacting topological insulators (TIs) via large-scale quantum Monte Carlo (QMC) simulations, based on the scheme of evaluating topological invariants presented in paper I. Across these transitions, the defining symmetries of the TIs have been neither explicitly nor spontaneously broken. In the first two models, the topological invariants calculated from the Green's function formalism succeed in characterizing the topologically distinct phases and identifying interaction-driven TPTs. However, in the other two models, we find that the single-particle gap does not close and the topological invariants constructed from the single-particle Green's function acquire no change across the TPTs. Unexpected breakdown of the Green's function formalism in constructing the topological invariants is thus discovered. We thence classify the topological phase transitions in interacting TIs into two categories in practical computation: Those that have noninteracting correspondence can be characterized successfully by the topological invariants constructed from the Green's functions, while for the others that do not have noninteracting correspondence, the Green's function formalism experiences a breakdown, but more interesting and exciting phenomena, such as emergent collective critical modes at the transition, arise. Discussion on the success and breakdown of topological invariants constructed from the Green's function formalism in the context of symmetry protected topological (SPT) states is presented.
We study a generalized Kane-Mele-Hubbard model with third-neighbor hopping, an interacting two-dimensional model with a topological phase transition as a function of third-neighbor hopping, by means of the determinant projector quantum Monte Carlo method. This technique is essentially numerically exact on models without a fermion sign problem, such as the one we consider. We determine the interaction dependence of the Z2{Z}_{2}Z2​ topological insulator/trivial insulator phase boundary by calculating the Z2{Z}_{2}Z2​ invariants directly from the single-particle Green's function. The interactions push the phase boundary to larger values of third-neighbor hopping, thus, stabilizing the topological phase. The observation of boundary shifting entirely stems from quantum fluctuations. We also identify qualitative features of the single-particle Green's function which are computationally useful in numerical searches for topological phase transitions without the need to compute the full topological invariant.
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.