Computing the ${\mathbb{Z}}_2$ Invariant in Two-Dimensional Strongly-Correlated Systems

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 wavefunction for both non-interacting 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 [Lee et al., Phys. Rev. Lett. $\text{100}$, 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.

v2: accepted version. 23 pages, 2 figures

References in corpus (29)

• Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells
• Topological Insulators with Inversion Symmetry
• Topological Field Theory of Time-Reversal Invariant Insulators
• First direct observation of Spin-textures in Topological Insulators : Spin-resolved ARPES as a probe of topological quantum spin Hall effect and Berry's phase
• Time Reversal Polarization and a Z_2 Adiabatic Spin Pump
• Signatures of Fractional Quantum Anomalous Hall States in Twisted MoTe2 Bilayer
• Observation of Fractionally Quantized Anomalous Hall Effect
• Quantum Spin Hall Effect and Topologically Invariant Chern Numbers
• Quantum anomalous Hall effect from intertwined moiré bands
• Bulk Topological Invariants in Noninteracting Point Group Symmetric Insulators
• Z2Pack: Numerical Implementation of Hybrid Wannier Centers for Identifying Topological Materials
• Observation of integer and fractional quantum anomalous Hall effects in twisted bilayer MoTe2
• Robustness of the Spin-Chern number
• Magnetic Topological Quantum Chemistry
• Building Blocks of Topological Quantum Chemistry: Elementary Band Representations
• All Topological Bands of All Nonmagnetic Stoichiometric Materials
• Topological Materials Discovery from Crystal Symmetry
• Vortexability: A Unifying Criterion for Ideal Fractional Chern Insulators
• Spin-Resolved Topology and Partial Axion Angles in Three-Dimensional Insulators
• Purely rotational symmetry-protected topological crystalline insulator $\alpha$-Bi4Br4
• Ideal Chern bands are Landau levels in curved space
• Many-body generalization of the Z2 topological invariant for the quantum spin Hall effect
• Failure of Topological Invariants in Strongly Correlated Matter
• 1/4 is the new 1/2 when topology is intertwined with Mottness
• Connecting the Many-Body Chern Number to Luttinger's Theorem through Středa's Formula
• Ground state stability, symmetry, and degeneracy in Mott insulators with long range interactions
• Lecture Notes on Berry Phases and Topology
• Stability of fractional Chern insulators with a non-Landau level continuum limit
• Topological Mott insulator in the odd-integer filled Anderson lattice model with Hatsugai-Kohmoto interactions

Cited by in corpus (6)

• Topic Review: Hatsugai-Kohmoto models: Exactly solvable playground for Mottness and Non-Fermi Liquid
• Band structure picture for topology in strongly correlated systems with the ghost Gutzwiller ansatz
• Topological charge excitations and Green's function zeros in paramagnetic Mott insulators
• Incipient quantum spin Hall insulator under strong correlations
• The antiferromagnetic Chern insulator phase in the Kane-Mele-Hubbard model
• Quantized and nonquantized Hall response in topological Hatsugai-Kohmoto systems