Mode-Shell correspondence, a unifying phase space theory in topological physics -- part II: Higher-dimensional spectral invariants

Abstract

The mode-shell correspondence relates the number $I_M$ of gapless modes in phase space to a topological \textit{shell invariant} $\Ish$ defined on a close surface -- the shell -- surrounding those modes, namely $I_M=I_S$. In part I, we introduced the mode-shell correspondence for zero-modes of chiral symmetric Hamiltonians (class AIII). In this part II, we extend the correspondence to arbitrary dimension and to both symmetry classes A and AIII. This allows us to include, in particular, $1D$-unidirectional edge modes of Chern insulators, massless $2D$-Dirac and $3D$-Weyl cones, within the same formalism. We provide an expression of $I_M$ that only depends on the dimension of the dispersion relation of the gapless mode, and does not require a translation invariance. Then, we show that the topology of the shell (a circle, a sphere, a torus), that must account for the spreading of the gapless mode in phase space, yields specific expressions of the shell index. Semi-classical expressions of those shell indices are also derived and reduce to either Chern or winding numbers depending on the parity of the mode's dimension. In that way, the mode-shell correspondence provides a unified and systematic topological description of both bulk and boundary gapless modes in any dimension, and in particular includes the bulk-boundary correspondence. We illustrate the generality of the theory by analyzing several models of semimetals and insulators, both on lattices and in the continuum, and also discuss weak and higher-order topological phases within this framework. Although this paper is a continuation of Part I, the content remains sufficiently independent to be mostly read separately.

References in corpus (64)

• Topological Insulators
• Electronic Structure of Pyrochlore Iridates: From Topological Dirac Metal to Mott Insulator
• Topological Insulators in Three Dimensions
• 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
• Room-Temperature Quantum Hall Effect in Graphene
• Topological insulators and superconductors: ten-fold way and dimensional hierarchy
• Quantized Electric Multipole Insulators
• Periodic table for topological insulators and superconductors
• Higher-Order Topological Insulators
• Chern Numbers in Discretized Brillouin Zone: Efficient Method of Computing (Spin) Hall Conductances
• Analogs of quantum Hall effect edge states in photonic crystals
• Topological response in Weyl semimetals and the chiral anomaly
• Topological quantization of the spin Hall effect in two-dimensional paramagnetic semiconductors
• Topological Boundary Modes in Isostatic Lattices
• The Zak phase and the existence of edge states in graphene
• Photonic topological pumping through the edges of a dynamical four-dimensional quantum Hall system
• Exploring 4D Quantum Hall Physics with a 2D Topological Charge Pump
• The Rare Two-Dimensional Materials with Dirac Cones
• Synthetic dimensions in integrated photonics: From optical isolation to 4D quantum Hall physics
• Chiral anomaly and transport in Weyl metals
• Photocurrents in Weyl semimetals
• Bulk-boundary correspondence of topological insulators from their Green's functions
• Bulk and Boundary Invariants for Complex Topological Insulators: From K-Theory to Physics
• Bulk-edge correspondence for two-dimensional topological insulators
• Topological Origin of Equatorial Waves
• The strong side of weak topological insulators
• Anomaly related transport of Weyl fermions for Weyl semi-metals
• Remarks on the tight-binding model of graphene
• Discovery of switchable weak topological insulator state in quasi-one-dimensional bismuth iodide
• Topological index for periodically driven time-reversal invariant 2D systems
• Quantum transport and two-parameter scaling at the surface of a weak topological insulator
• Geometrical engineering of a two-bands Chern insulator in two dimensions with arbitrary topological index
• Simultaneous quantization of edge and bulk Hall conductivity
• Disordered weak and strong topological insulators
• Topological and geometrical aspects of band theory
• Type-II Weyl cone transitions in driven semimetals
• Introduction to Dirac materials and topological insulators
• Topological Weyl Semi-metal from a Lattice Model
• Aperiodic Weak Topological Superconductors
• Prediction of weak topological insulators in layered semiconductors
• Observation and control of the weak topological insulator state in ZrTe5
• Higher-order topological phases on fractal lattices
• Topological zero modes and Dirac points protected by spatial symmetry and chiral symmetry
• A bulk-interface correspondence for equatorial waves
• The Bulk-Edge Correspondence for Disordered Chiral Chains
• Topology in shallow-water waves: a violation of bulk-edge correspondence
• Experimental realization of two-dimensional weak topological insulators
• Topological Langmuir-cyclotron wave
• Index theorem on chiral Landau bands for topological fermions
• Hidden symmetry and protection of Dirac points on the honeycomb lattice
• Wave topology brought to the coast
• Chiral Maxwell waves in continuous media from Berry monopoles
• Estimating bulk and edge topological indices in finite open chiral chains
• Geometry and Topology Tango in Ordered and Amorphous Chiral Matter
• Non-Hermitian spectral flows and Berry-Chern monopoles
• Generalization of Benalcazar-Bernevig-Hughes model to arbitrary dimensions
• Tunneling in the Brillouin Zone: Theory of Backscattering in Valley Hall Edge Channels
• Topological modes in stellar oscillations
• Tunable Dirac points in a two-dimensional non-symmorphic wallpaper group lattice
• Pairwise annihilation of Weyl nodes induced by magnetic fields in the Hofstadter regime
• Change of the Chern number at band crossings
• Multi-type Dirac fermions protected by orthogonal glide symmetries in a noncentrosymmetric system

Cited by in corpus (1)

• Phase-Space Topology in a Single-Atom Synthetic Dimension