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.

26 pages + appendices, 12 figures

References in corpus (22)

• 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 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
• Electric Multipole Moments, Topological Multipole Moment Pumping, and Chiral Hinge States in Crystalline Insulators
• Topological Defects and Gapless Modes in Insulators and Superconductors
• 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
• Disclinations, dislocations and continuous defects: a reappraisal
• Axion topological field theory of topological superconductors
• 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
• Topological Zero-Dimensional Defect and Flux States in Three-Dimensional Insulators
• Fractional disclination charge and discrete shift in the Hofstadter butterfly

Cited by in corpus (1)

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