Interacting Topological Quantum Chemistry in 2D: Many-body Real Space Invariants

Abstract

The topological phases of non-interacting fermions have been classified by their symmetries, culminating in a modern electronic band theory where wavefunction topology can be obtained (in part) from momentum space. Recently, Real Space Invariants (RSIs) have provided a spatially local description of the global momentum space indices. The present work generalizes this real space classification to interacting 2D states. We construct many-body local RSIs as the quantum numbers of a set of symmetry operators on open boundaries, but which are independent of the choice of boundary. Using the $U(1)$ particle number, they yield many-body fragile topological indices, which we use to identify which single-particle fragile states are many-body topological or trivial at weak coupling. To this end, we construct an exactly solvable Hamiltonian with single-particle fragile topology that is adiabatically connected to a trivial state through strong coupling. We then define global many-body RSIs on periodic boundary conditions. They reduce to Chern numbers in the band theory limit, but also identify strongly correlated stable topological phases with no single-particle counterpart. Finally, we show that the many-body local RSIs appear as quantized coefficients of Wen-Zee terms in the topological quantum field theory describing the phase.

arXiv comment5 + 36 pages

References in corpus (91)

• Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells
• $Z_2$ Topological Order and the Quantum Spin Hall Effect
• Topological Field Theory of Time-Reversal Invariant Insulators
• Maximally localized Wannier functions: Theory and applications
• Classification of topological quantum matter with symmetries
• Topological insulators and superconductors: ten-fold way and dimensional hierarchy
• Quantized Electric Multipole Insulators
• Periodic table for topological insulators and superconductors
• Topological quantum chemistry
• Electric Multipole Moments, Topological Multipole Moment Pumping, and Chiral Hinge States in Crystalline Insulators
• $(d-2)$-dimensional edge states of rotation symmetry protected topological states
• 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
• Symmetry-based Indicators of Band Topology in the 230 Space Groups
• Superfluidity in topologically nontrivial flat bands
• The effects of interactions on the topological classification of free fermion systems
• Inversion Symmetric Topological Insulators
• Bulk Topological Invariants in Noninteracting Point Group Symmetric Insulators
• All "Magic Angles" Are "Stable" Topological
• Theory and classification of interacting 'integer' topological phases in two dimensions: A Chern-Simons approach
• Wannier representation of Z_2 topological insulators
• Exponential localization of Wannier functions in insulators
• Faithful Tight-binding Models and Fragile Topology of Magic-angle Bilayer Graphene
• Fragile Topology and Wannier Obstructions
• Interacting topological insulators: a review
• Single particle Green's functions and interacting topological insulators
• Quantitative mappings between symmetry and topology in solids
• Bulk-boundary correspondence of topological insulators from their Green's functions
• Magnetic Topological Quantum Chemistry
• Topology-Bounded Superfluid Weight In Twisted Bilayer Graphene
• Structure and Topology of Band Structures in the 1651 Magnetic Space Groups
• Wilson loop approach to fragile topology of split elementary band representations and topological crystalline insulators with time reversal symmetry
• Exact Landau Level Description of Geometry and Interaction in a Flatband
• Classifying gauge anomalies through SPT orders and classifying gravitational anomalies through topological orders
• Fractional disclination charge in two-dimensional $C_n-$symmetric topological crystalline insulators
• Electromagnetic and gravitational responses of two-dimensional non-interacting electrons in background magnetic field
• Real Space Invariants: Twisted Bulk-Boundary Correspondence of Fragile Topology
• Twisted Bilayer Graphene V: Exact Analytic Many-Body Excitations in Twisted Bilayer Graphene Coulomb Hamiltonians: Charge Gap, Goldstone Modes and Absence of Cooper Pairing
• Topology of Disconnected Elementary Band Representations
• Experimental characterization of fragile topology in an acoustic metamaterial
• Fragile topology and flat-band superconductivity in the strong-coupling regime
• Low-energy effective theory in the bulk for transport in a topological phase
• Band Representations and Topological Quantum Chemistry
• Superfluid weight bounds from symmetry and quantum geometry in flat bands
• Theory of interacting topological crystalline insulators
• Many-body topological invariants for fermionic symmetry-protected topological phases
• Many-body topological invariants for fermionic short-range entangled topological phases protected by antiunitary symmetries
• Quantized topological terms in weak-coupling gauge theories with symmetry and their connection to symmetry enriched topological phases
• Hofstadter Topology: Non-crystalline Topological Materials at High Flux
• Topological central charge from Berry curvature: gravitational anomalies in trial wavefunctions for topological phases
• Entanglement Spectrum Classification of $C_n$-invariant Noninteracting Topological Insulators in Two Dimensions
• Many-body topological invariants in fermionic symmetry protected topological phases: Cases of point group symmetries
• Optical Spectral Weight, Phase Stiffness and Tc Bounds for Trivial and Topological Flat Band Superconductors
• Many-body Chern number from statistical correlations of randomized measurements
• Fragile Phases As Affine Monoids: Classification and Material Examples
• Symmetry indicators of band topology
• Classification of (2+1)D invertible fermionic topological phases with symmetry
• Topological Semimetal driven by Strong Correlations and Crystalline Symmetry
• Fragile Mott Insulators
• Many-body generalization of the Z2 topological invariant for the quantum spin Hall effect
• Crystalline gauge fields and quantized discrete geometric response for Abelian topological phases with lattice symmetry
• Boundary conformal field theory and symmetry protected topological phases in $2+1$ dimensions
• Reentrant Correlated Insulators in Twisted Bilayer Graphene at 25T ($2\pi$ Flux)
• Fragile topological phases in interacting systems
• Higher-order topological pumping and its observation in photonic lattices
• Extraction of many-body Chern number from a single wave function
• Pairing Obstructions in Topological Superconductors
• Hofstadter Topology with Real Space Invariants and Reentrant Projective Symmetries
• Real-space construction of crystalline topological superconductors and insulators in 2D interacting fermionic systems
• Fractional disclination charge and discrete shift in the Hofstadter butterfly
• Diamagnetic response and phase stiffness for interacting isolated narrow bands
• Quantized charge polarization as a many-body invariant in (2+1)D crystalline topological states and Hofstadter butterflies
• Interaction Enabled Topological Crystalline Phases
• Interacting topological quantum chemistry of Mott atomic limits
• Elementary band representations for the single-particle Green's function of interacting topological insulators
• Non-Compact Atomic Insulators
• Three-Dimensional Real Space Invariants, Obstructed Atomic Insulators and A New Principle for Active Catalytic Sites
• The Mott transition as a topological phase transition
• Universal Relation among Many-Body Chern Number, Rotation Symmetry, and Filling
• Crystalline Responses for Rotation-Invariant Higher-Order Topological Insulators
• Correlated Fragile Topology: a Parton Approach
• Quantum metric on the Brillouin Zone in correlated electron systems and its relation to topology for Chern insulators
• Drude weight and the many-body quantum metric in one-dimensional Bose systems
• Effective field theories of topological crystalline insulators and topological crystals
• Effective action approach to the filling anomaly in crystalline topological matter
• Topological Green function of interacting systems
• Zeros of Green Functions in Topological Insulators
• Non-local order parameters for states with topological electromagnetic response
• Gravitational Response of Topological Quantum States of Matter
• Construction of interacting flat-band models by molecular-orbital representation: Correlation functions, energy gap, and entanglement
• Exactly solvable lattice models for interacting electronic insulators in two dimensions

Cited by in corpus (13)

• Mott insulators with boundary zeros
• Moiré Fractional Chern Insulators III: Hartree-Fock Phase Diagram, Magic Angle Regime for Chern Insulator States, the Role of the Moiré Potential and Goldstone Gaps in Rhombohedral Graphene Superlattices
• "Quantum Geometric Nesting" and Solvable Model Flat-Band Systems
• Tutorial: Classifying Photonic Topology Using the Spectral Localizer and Numerical $K$-Theory
• Interaction-induced crystalline topology of excitons
• Altermagnetic Instabilities from Quantum Geometry
• Exciton Berryology
• Electric polarization and discrete shift from boundary and corner charge in crystalline Chern insulators
• Crystalline invariants of fractional Chern insulators
• Interacting Crystalline Topological Insulators in two-dimensions with Time-Reversal Symmetry
• Global and Local Topological Crystalline Markers for Rotation-Symmetric Insulators
• (2+1)D topological phases with RT symmetry: many-body invariant, classification, and higher order edge modes
• Berry Curvature of Low-Energy Excitons in Rhombohedral Graphene