Unveiling Correlated Two-dimensional Topological Insulators through Fermionic Tensor Network States -- Classification, Edge Theories and Variational Wavefunctions

Abstract

The study of topological band insulators has revealed fascinating phases characterized by band topology indices and anomalous boundary modes protected by global symmetries. In strongly correlated systems, where the traditional notion of electronic bands becomes obsolete, it has been established that the topological insulator phases persist as stable phases, separate from the trivial insulators. However, due to the inability to express the ground states of such systems as Slater determinants, the formulation of generic variational wavefunctions for numerical simulations is highly desirable. In this paper, we tackle this challenge for two-dimensional topological insulators by developing a comprehensive framework for fermionic tensor network states. Starting from simple assumptions, we obtain possible sets of tensor equations for any given symmetry group, capturing consistent relations governing symmetry transformation rules on tensor legs. We then examine the connections between these tensor equations and non-chiral topological insulators by construing edge theories and extracting quantum anomaly data from each set of tensor equations. By exhaustively exploring all possible sets of equations, we achieve a systematic classification of non-chiral topological insulator phases. Imposing the solutions of a given set of equations onto local tensors, we obtain generic variational wavefunctions for corresponding topological insulator phases. Our methodology provides an important step towards simulating topological insulators in strongly correlated systems. We discuss the limitations and potential generalizations of our results, paving the way for further advancements in this field.

32+20 pages, 11 figures

References in corpus (88)

• Topological Insulators
• Topological insulators and superconductors
• Quantum Spin Hall Effect in Graphene
• $Z_2$ Topological Order and the Quantum Spin Hall Effect
• Quantum Spin Hall Insulator State in HgTe Quantum Wells
• Topological Insulators in Three Dimensions
• Topological Insulators with Inversion Symmetry
• Classification of topological insulators and superconductors in three spatial dimensions
• Topological Field Theory of Time-Reversal Invariant Insulators
• Classification of topological quantum matter with symmetries
• Periodic table for topological insulators and superconductors
• Higher-Order Topological Insulators
• Quantum Orders and Symmetric Spin Liquids
• Classification of Gapped Symmetric Phases in 1D Spin Systems
• Topological phases of fermions in one dimension
• Matrix Product States and Projected Entangled Pair States: Concepts, Symmetries, and Theorems
• Classifying quantum phases using Matrix Product States and PEPS
• Symmetry-based Indicators of Band Topology in the 230 Space Groups
• Towards ideal topological materials: Comprehensive database searches using symmetry indicators
• The Helical Liquid and the Edge of Quantum Spin Hall Systems
• Topological classification of crystalline insulators through band structure combinatorics
• Tensor renormalization group approach to 2D classical lattice models
• Braiding statistics approach to symmetry-protected topological phases
• The effects of interactions on the topological classification of free fermion systems
• Stability of the quantum spin Hall effect: effects of interactions, disorder, and Z_2 topology
• Topological Phases of One-Dimensional Fermions: An Entanglement Point of View
• Physics of three dimensional bosonic topological insulators: Surface Deconfined Criticality and Quantized Magnetoelectric Effect
• Theory and classification of interacting 'integer' topological phases in two dimensions: A Chern-Simons approach
• Complete classification of 1D gapped quantum phases in interacting spin systems
• Coarse-graining renormalization by higher-order singular value decomposition
• Is the Composite Fermion a Dirac Particle?
• Simulation of two dimensional quantum systems on an infinite lattice revisited: corner transfer matrix for tensor contraction
• Fermionic Symmetry Protected Topological Phases and Cobordisms
• Detection of Symmetry Protected Topological Phases in 1D
• Fractional topological insulators
• Quantitative mappings between symmetry and topology in solids
• Symmetry-protected topological orders for interacting fermions -- Fermionic topological nonlinear $\sigma$ models and a special group supercohomology theory
• PEPS as ground states: degeneracy and topology
• Interacting fermionic topological insulators/superconductors in three dimensions
• Tensor-entanglement renormalization group approach to 2D quantum systems
• Higher-Order Topological Insulator in Twisted Bilayer Graphene
• Classification of interacting electronic topological insulators in three dimensions
• Fermionic Projected Entangled Pair States
• Higher-Order Topological Insulators in Quasicrystals
• Differentiable Programming Tensor Networks
• Symmetry protected topological phases from decorated domain walls
• Integer quantum Hall effect for bosons: A physical realization
• Particle-vortex duality of 2d Dirac fermion from electric-magnetic duality of 3d topological insulators
• Classifying symmetry-protected topological phases through the anomalous action of the symmetry on the edge
• Choreographed entangle dances: topological states of quantum matter
• Fractional topological liquids with time-reversal symmetry and their lattice realization
• Dual Dirac liquid on the surface of the electron topological insulator
• Equivalence of the Variational Matrix Product Method and the Density Matrix Renormalization Group applied to Spin Chains
• Loop optimization for tensor network renormalization
• Spin TQFTs and fermionic phases of matter
• Intrinsically Gapless Topological Phases
• Contraction of fermionic operator circuits and the simulation of strongly correlated fermions
• Simulation of interacting fermions with entanglement renormalization
• Tensor network renormalization yields the multi-scale entanglement renormalization ansatz
• Fermionic Matrix Product States and One-Dimensional Topological Phases
• Particle-Hole Symmetry and the Composite Fermi Liquid
• An order parameter for symmetry-protected phases in one dimension
• Half-filled Landau level, topological insulator surfaces, and three dimensional quantum spin liquids
• Towards a complete classification of fermionic symmetry protected topological phases in 3D and a general group supercohomology theory
• Dimers on surface graphs and spin structures. I
• Crystalline topological phases as defect networks
• Construction and classification of symmetry protected topological phases in interacting fermion systems
• Topological Response Theory of Abelian Symmetry-Protected Topological Phases in Two Dimensions
• Classification and construction of higher-order symmetry protected topological phases of interacting bosons
• Classification of (2+1)D invertible fermionic topological phases with symmetry
• Discrete spin structures and commuting projector models for 2d fermionic symmetry protected topological phases
• Bulk-boundary correspondence for intrinsically-gapless SPTs from group cohomology
• Anyon condensation and a generic tensor-network construction for symmetry protected topological phases
• Real-space recipes for general topological crystalline states
• Symmetric tensor networks and practical simulation algorithms to sharply identify classes of quantum phases distinguishable by short-range physics
• Fermionic projected entangled-pair states and topological phases
• Symmetry-protected topological phases with charge and spin symmetries: response theory and dynamical gauge theory in 2D, 3D and the surface of 3D
• ${\mathbb{Z}}_2$ fractional topological insulators in two dimensions
• Tensor network representations of parton wave functions
• Ising Anyons in Frustration-Free Majorana-Dimer Models
• Microscopic Realization of 2-Dimensional Bosonic Topological Insulators
• Fermionic topological quantum states as tensor networks
• Interaction Driven Quantum Phase Transitions in Fractional Topological Insulators
• Grassmann tensor network states and its renormalization for strongly correlated fermionic and bosonic states
• Tensor Networks Can Resolve Fermi Surfaces
• Disentangling interacting symmetry protected phases of fermions in two dimensions
• Exactly Solvable Model for Two Dimensional Topological Superconductor
• Commuting-projector Hamiltonians for 2D topological insulators: edge physics and many-body invariants

Cited by in corpus (4)

• Fermionic tensor network methods
• Band structure picture for topology in strongly correlated systems with the ghost Gutzwiller ansatz
• Boundary anomaly detection in two-dimensional subsystem symmetry-protected topological phases
• Frustration-free free fermions