Mode-Shell correspondence, a unifying phase space theory in topological physics -- Part I: Chiral number of zero-modes
arXiv:2310.05656 · doi:10.21468/SciPostPhys.17.2.060
Abstract
We propose a theory, that we call the \textit{mode-shell correspondence}, which relates the topological zero-modes localised in phase space to a \textit{shell} invariant defined on the surface forming a shell enclosing these zero-modes. We show that the mode-shell formalism provides a general framework unifying important results of topological physics, such as the bulk-edge correspondence, higher-order topological insulators, but also the Atiyah-Singer and the Callias index theories. In this paper, we discuss the already rich phenomenology of chiral symmetric Hamiltonians where the topological quantity is the chiral number of zero-dimensionial zero-energy modes. We explain how, in a lot of cases, the shell-invariant has a semi-classical limit expressed as a generalised winding number on the shell, which makes it accessible to analytical computations.
74 pages, 20 figures
References in corpus (25)
- 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
- Reflection-Free One-Way Edge Modes in a Gyromagnetic Photonic Crystal
- Electric Multipole Moments, Topological Multipole Moment Pumping, and Chiral Hinge States in Crystalline Insulators
- Analogs of quantum Hall effect edge states in photonic crystals
- Topological characterization of periodically-driven quantum systems
- -dimensional edge states of rotation symmetry protected topological states
- Reflection symmetric second-order topological insulators and superconductors
- Topological Defects and Gapless Modes in Insulators and Superconductors
- Quantum Spin Hall Effect and Topologically Invariant Chern Numbers
- Exploring Topological Phases With Quantum Walks
- Selective enhancement of topologically induced interface states in a dielectric resonator chain
- New topological invariants in non-Hermitian systems
- Chiral symmetry and bulk--boundary correspondence in periodically driven one-dimensional systems
- Non-Bloch topological invariants in a non-Hermitian domain-wall system
- Topological index for periodically driven time-reversal invariant 2D systems
- Gain and loss induced topological insulating phase in a non Hermitian electrical circuit
- Topological Weyl Semi-metal from a Lattice Model
- Prediction of weak topological insulators in layered semiconductors
- Observation and control of the weak topological insulator state in ZrTe5
- Higher-order topological phases in crystalline and non-crystalline systems: a review
- Experimental realization of two-dimensional weak topological insulators
- Multiplicative topological phases
- Topological modes in stellar oscillations
- The Analysis of Bulk Boundary Correspondence under the Singularity of the Generalized Brillouin Zone in Non-Hermitian System