Shift insulators: rotation-protected two-dimensional topological crystalline insulators

Abstract

We study a two-dimensional (2D) tight-binding model of a topological crystalline insulator (TCI) protected by rotation symmetry. The model is built by stacking two Chern insulators with opposite Chern numbers which transform under conjugate representations of the rotation group, e.g. $p_{\pm }$ orbitals. Despite its apparent similarity to the Kane-Mele model, it does not host stable gapless surface states. Nevertheless the model exhibits topological responses including the appearance of quantized fractional charge bound to rotational defects (disclinations) and the pumping of angular momentum in response to threading an elementary magnetic flux, which are described by a mutual Chern-Simons coupling between the electromagnetic gauge field and an effective gauge field corresponding to the rotation symmetry. In addition, we show that although the filled bands of the model do not admit a symmetric Wannier representation, this obstruction is removed upon the addition of appropriate atomic orbitals, which implies `fragile' topology. As a result, the response of the model can be derived by representing it as a superposition of atomic orbitals with positive and negative integer coefficients. Following the analysis of the model, which serves as a prototypical example of 2D TCIs protected by rotation, we show that all TCIs protected by point group symmetries which do not have protected surface states are either atomic insulators or fragile phases. Remarkably, this implies that gapless surface states exist in free electron systems if and only if there is a stable Wannier obstruction. We then use dimensional reduction to map the problem of classifying 2D TCIs protected by rotation to a zero-dimensional (0D) problem which is then used to obtain the complete non-interacting classification of such TCIs as well as the reduction of this classification in the presence of interactions.

arXiv comment33 pages, 16 figures

References in corpus (64)

• Topological Insulators
• Topological insulators and superconductors
• Quantum Spin Hall Effect in Graphene
• $Z_2$ Topological Order and the Quantum Spin Hall Effect
• Maximally-localized generalized Wannier functions for composite energy bands
• 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
• Topological Crystalline Insulators
• Quantized Electric Multipole Insulators
• Periodic table for topological insulators and superconductors
• Higher-Order Topological Insulators
• Topological quantum chemistry
• Topological Crystalline Insulators in the SnTe Material Class
• 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
• Experimental realization of a topological crystalline insulator in SnTe
• Reflection symmetric second-order topological insulators and superconductors
• Time Reversal Polarization and a Z_2 Adiabatic Spin Pump
• Topological Defects and Gapless Modes in Insulators and Superconductors
• Symmetry-based Indicators of Band Topology in the 230 Space Groups
• Topological classification of crystalline insulators through band structure combinatorics
• Higher-order topological insulators and superconductors protected by inversion symmetry
• Topology of crystalline insulators and superconductors
• Bulk Topological Invariants in Noninteracting Point Group Symmetric Insulators
• Boundary conditions for Dirac fermions on a terminated honeycomb lattice
• Wannier representation of Z_2 topological insulators
• Exponential localization of Wannier functions in insulators
• Second-order topological insulators and superconductors with an order-two crystalline symmetry
• Fragile Topology and Wannier Obstructions
• Classification of reflection symmetry protected topological semimetals and nodal superconductors
• Quantitative mappings between symmetry and topology in solids
• Classification of Topological Insulators and Superconductors
• Higher-order bulk-boundary correspondence for topological crystalline phases
• Symmetry indicators and anomalous surface states of topological crystalline insulators
• Quantized Response and Topology of Insulators with Inversion Symmetry
• Topological classification with additional symmetries from Clifford algebras
• Structure and Topology of Band Structures in the 1651 Magnetic Space Groups
• Higher-order topological insulators protected by inversion and rotoinversion symmetries
• Classification of Two Dimensional Topological Crystalline Superconductors and Majorana Bound States at Disclinations
• Topological phases protected by point group symmetry
• Classification and analysis of two dimensional abelian fractional topological insulators
• Building crystalline topological phases from lower-dimensional states
• The insulator/Chern-insulator transition in the Haldane model
• Stability of Chiral Luttinger Liquids and Abelian Quantum Hall States.
• Gauging spatial symmetries and the classification of topological crystalline phases
• Bulk-boundary correspondence from the inter-cellular Zak phase
• The effect of interactions on 2D fermionic symmetry-protected topological phases with Z2 symmetry
• Topology of Disconnected Elementary Band Representations
• Topological insulators in magnetic fields: Quantum Hall effect and edge channels with non-quantized θ-term
• Topological Phases in Graphitic Cones
• Spin-charge Separated Solitons in a Topological Band Insulator
• Theory of interacting topological crystalline insulators
• Diagnosis for topological semimetals in the absence of spin-orbital coupling
• Lattice Homotopy Constraints on Phases of Quantum Magnets
• Topological superfluid 3He-B in magnetic field and Ising variable
• Bound States of Conical Singularities in Graphene-Based Topological Insulators
• Fragile topological phases in interacting systems
• Hourglass Fermion Surface States in Stacked Topological Insulators with Nonsymmorphic Symmetry
• Coupled-layer description of topological crystalline insulators
• Corner states of topological fullerenes
• No-go theorem for topological insulators and sure-fire recipe for Chern insulators
• Fullerenes, Zero-modes, and Self-adjoint Extensions

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• Generic "Unnecessary" Quantum Critical Points with Minimal Degrees of Freedom
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• Dynamical Fragile Topology in Floquet Crystals
• Interacting Valley Chern Insulator and its Topological Imprint on Moiré Superconductors
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