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arXiv:1201.6431 · Physical Review B

Topological invariants for interacting topological insulators with inversion symmetry

Zhong Wang, Xiao-Liang Qi, Shoucheng Zhang

Published Apr 17, 2012• 87 citations
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Abstract

For interacting ${Z}_{2}$ 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 ${Z}_{2}$ topological invariants in inversion-symmetric insulators with strong interactions.

DOI
10.1103/physrevb.85.165126

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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.
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.
We use a ``monodromy'' argument to derive new expressions for the Z2{\mathbb{Z}}_{2}Z2​ invariants of topological insulators with time-reversal symmetry in two and three dimensions. The derivations and the final expressions do not require any gauge choice and the calculation of the invariants is based entirely on the projectors onto the occupied states. Explicit numerical tests for tight-binding models with strongly broken inversion symmetry are presented in two and three dimensions.
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.
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.
We study the effect of interactions on the time-reversal invariant topological insulators in four and three spatial dimensions. Their topological indices are expressed by the interacting Green's functions. Under the local self-energy approximation, we find that interaction could induce nontrivial frequency-domain winding numbers and change the topological classes of the system. Our results suggest that the topological phases could be destroyed without developing long-range orders. Practical issues on the accurate frequency-momentum integration combined with dynamical-mean-field theory and diagrammatic calculations of the interacting Green's functions are also addressed.
The Bloch wave functions have been playing a crucial role in the diagnosis of topological phases in noninteracting systems. However, the Bloch waves are no longer applicable in the presence of finite Coulomb interaction and alternative approaches are needed to identify the topological indices. In this paper, we focus on three-dimensional higher-order topological insulators protected by C4T{C}_{4}TC4​T symmetry and show that the topological index can be computed through eigenstates of inverse Green's function at zero frequency. If there is an additional S4{S}_{4}S4​ rotoinversion symmetry, the topological index P3{P}_{3}P3​ can be determined by eigenvalues of S4{S}_{4}S4​ at high-symmetry momenta, similar to the Fu-Kane parity criterion. We verify this method using many-body exact diagonalization in higher-order topological insulators with interaction. We also discuss the realization of this higher-order topological phase in tetragonal lattice structure with C4T{C}_{4}TC4​T-preserving magnetic order. Finally, we discuss the boundary conditions necessary for the hinge states to emerge and show that these hinge states exist even when the boundary is smooth and without a sharp hinge.