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 (66)

• Topological Insulators
• Magic-angle graphene superlattices: a new platform for unconventional superconductivity
• Quantum Spin Hall Effect in Graphene
• Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells
• $Z_2$ Topological Order and the Quantum Spin Hall Effect
• Anyons in an exactly solved model and beyond
• Topological Insulators in Three Dimensions
• Topological Insulators with Inversion Symmetry
• Topological Field Theory of Time-Reversal Invariant Insulators
• Classification of topological quantum matter with symmetries
• Topological invariants of time-reversal-invariant band structures
• Topological quantum chemistry
• Topological Crystalline Insulators in the SnTe Material Class
• Chern Numbers in Discretized Brillouin Zone: Efficient Method of Computing (Spin) Hall Conductances
• Beyond Dirac and Weyl fermions: Unconventional quasiparticles in conventional crystals
• 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
• The (High Quality) Topological Materials In The World
• Magic continuum in twisted bilayer WSe2
• Time Reversal Polarization and a Z_2 Adiabatic Spin Pump
• An equivalent expression of Z2 Topological Invariant for band insulators using Non-Abelian Berry's connection
• Zoo of quantum-topological phases of matter
• Symmetry-based Indicators of Band Topology in the 230 Space Groups
• Towards ideal topological materials: Comprehensive database searches using symmetry indicators
• Signatures of Fractional Quantum Anomalous Hall States in Twisted MoTe2 Bilayer
• A topological crystalline insulator (TCI) phase via topological phase transition and crystalline mirror symmetry
• Observation of Fractionally Quantized Anomalous Hall Effect
• Quantum Spin Hall Effect and Topologically Invariant Chern Numbers
• Topological classification of crystalline insulators through band structure combinatorics
• Quantum anomalous Hall effect from intertwined moiré bands
• Symmetry Fractionalization, Defects, and Gauging of Topological Phases
• Bulk Topological Invariants in Noninteracting Point Group Symmetric Insulators
• Z2Pack: Numerical Implementation of Hybrid Wannier Centers for Identifying Topological Materials
• Fermion Path Integrals And Topological Phases
• Observation of integer and fractional quantum anomalous Hall effects in twisted bilayer MoTe2
• Single particle Green's functions and interacting topological insulators
• Robustness of the Spin-Chern number
• Polarization and localization in insulators: generating function approach
• Quantitative mappings between symmetry and topology in solids
• Band geometry of fractional topological insulators
• Symmetry indicators and anomalous surface states of topological crystalline insulators
• Magnetic Topological Quantum Chemistry
• Building Blocks of Topological Quantum Chemistry: Elementary Band Representations
• Twisted equivariant matter
• Chiral Optical Response of Multifold Fermions
• All Topological Bands of All Nonmagnetic Stoichiometric Materials
• Topological Order Parameters for Interacting Topological Insulators
• Efficient Topological Materials Discovery Using Symmetry Indicators
• Topological Materials Discovery from Crystal Symmetry
• Zoology of Fractional Chern Insulators
• Building crystalline topological phases from lower-dimensional states
• Gauging spatial symmetries and the classification of topological crystalline phases
• Vortexability: A Unifying Criterion for Ideal Fractional Chern Insulators
• Band Representations and Topological Quantum Chemistry
• Geometric stability of topological lattice phases
• Many-body topological invariants for fermionic short-range entangled topological phases protected by antiunitary symmetries
• Spin-Resolved Topology and Partial Axion Angles in Three-Dimensional Insulators
• Ideal Chern bands are Landau levels in curved space
• Purely rotational symmetry-protected topological crystalline insulator $\alpha$-Bi4Br4
• 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

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