Topological Field Theories of Three-Dimensional Rotation Symmetric Insulators: Coupling Curvature and Electromagnetism

Abstract

Quantized responses are important tools for understanding and characterizing the universal features of topological phases of matter. In this work, we consider a class of topological crystalline insulators in $3$D with $C_n$ lattice rotation symmetry along a fixed axis, in addition to either mirror symmetry or particle-hole symmetry. These insulators can realize quantized mixed geometry-charge responses. When the surface of these insulators is gapped, disclinations on the surface carry a fractional charge that is half the minimal amount that can occur in purely $2$D systems. Similarly, disclination lines in the bulk carry a fractionally quantized electric polarization. These effects, and other related phenomena, are captured by a $3$D topological response term that couples the lattice curvature to the electromagnetic field strength. Additionally, mirror symmetric insulators with this response can be smoothly deformed into a higher-order octopole insulator with quantized corner charges. We also construct a symmetry indicator form for the topological invariant that describes the quantized response of the mirror symmetric topological crystalline insulators, and discuss an unusual response quantization in time-reversal breaking systems.

arXiv comment26 pages + appendices, 12 figures

References in corpus (70)

• Topological Insulators
• Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells
• Quantum Spin Hall Insulator State in HgTe Quantum Wells
• Superconducting proximity effect and Majorana fermions at the surface of a topological insulator
• Topological Insulators in Three Dimensions
• Topological Insulators with Inversion Symmetry
• Discovery (theoretical prediction and experimental observation) of a large-gap topological-insulator class with spin-polarized single-Dirac-cone on the surface
• A topological Dirac insulator in a quantum spin Hall phase : Experimental observation of first strong topological insulator
• Classification of topological insulators and superconductors in three spatial dimensions
• Topological Field Theory of Time-Reversal Invariant Insulators
• Topological Crystalline Insulators
• Quantized Electric Multipole Insulators
• Higher-Order Topological Insulators
• Topological superconductors: a review
• Electric Multipole Moments, Topological Multipole Moment Pumping, and Chiral Hinge States in Crystalline Insulators
• Symmetry protected topological orders and the group cohomology of their symmetry group
• Tensor-Entanglement-Filtering Renormalization Approach and Symmetry Protected Topological Order
• Symmetry protection of topological order in one-dimensional quantum spin systems
• Topological Defects and Gapless Modes in Insulators and Superconductors
• Quantization of fractional corner charge in $C_n$-symmetric higher-order topological crystalline insulators
• Inversion Symmetric Topological Insulators
• Higher-order topological insulators and superconductors protected by inversion symmetry
• Dynamical Axion Field in Topological Magnetic Insulators
• Bulk Topological Invariants in Noninteracting Point Group Symmetric Insulators
• Physics of three dimensional bosonic topological insulators: Surface Deconfined Criticality and Quantized Magnetoelectric Effect
• Helical Metal Inside a Topological Band Insulator
• Electromagnetic and gravitational responses and anomalies in topological insulators and superconductors
• Interacting fermionic topological insulators/superconductors in three dimensions
• Disclinations, dislocations and continuous defects: a reappraisal
• Disclination Loops, Hedgehogs, and All That
• Bulk and Boundary Invariants for Complex Topological Insulators: From K-Theory to Physics
• Topological Nematic States and Non-Abelian Lattice Dislocations
• Integer quantum Hall effect for bosons: A physical realization
• Classification of Two Dimensional Topological Crystalline Superconductors and Majorana Bound States at Disclinations
• Topological phases protected by point group symmetry
• Theta-terms in nonlinear sigma-models
• Majorana Fermions and Disclinations in Topological Crystalline Superconductors
• Surfaces of Axion Insulators
• Fractional disclination charge in two-dimensional $C_n-$symmetric topological crystalline insulators
• Gapped Symmetry Preserving Surface-State for the Electron Topological Insulator
• A symmetry-respecting topologically-ordered surface phase of 3d electron topological insulators
• Bosonic topological insulator in three dimensions and the statistical Witten effect
• Torsional Response and Dissipationless Viscosity in Topological Insulators
• Boundary-obstructed topological phases
• Universal probes of two-dimensional topological insulators: Dislocation and pi-flux
• Classification and Description of Bosonic Symmetry Protected Topological Phases with semiclassical Nonlinear Sigma models
• Classification and Properties of Symmetry Enriched Topological Phases: A Chern-Simons approach with applications to Z2 spin liquids
• Axion topological field theory of topological superconductors
• Field theory of the quantum Hall nematic transition
• Self-dual Quantum Electrodynamics as Boundary State of the three dimensional Bosonic Topological Insulator
• Multipolar Topological Field Theories: Bridging Higher Order Topological Insulators and Fractons
• Shift insulators: rotation-protected two-dimensional topological crystalline insulators
• Theory of Nematic Fractional Quantum Hall State
• Topological indices, defects and Majorana fermion in chiral superconductors
• Irrational vs. rational charge and statistics in two-dimensional quantum systems
• Bound States of Conical Singularities in Graphene-Based Topological Insulators
• Classification of Crystalline Topological Insulators and Superconductors with Point Group Symmetries
• Anomaly indicators for time-reversal symmetric topological orders
• Dislocation as a bulk probe of higher-order topological insulators
• Crystalline gauge fields and quantized discrete geometric response for Abelian topological phases with lattice symmetry
• Topological Zero-Dimensional Defect and Flux States in Three-Dimensional Insulators
• Generalized Wen-Zee Terms
• Fractional disclination charge and discrete shift in the Hofstadter butterfly
• Corner states of topological fullerenes
• Disclination classes, fractional excitations, and the melting of quantum liquid crystals
• Unquantized anomalies in topological semimetals
• Crystalline Responses for Rotation-Invariant Higher-Order Topological Insulators
• Effective field theories of topological crystalline insulators and topological crystals
• Topological Dipole Conserving Insulators and Multipolar Responses
• Electromagnetic Response of Three-dimensional Topological Crystalline Insulators

Cited by in corpus (1)

• Effective action approach to the filling anomaly in crystalline topological matter