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

TL;DR

This paper extends the mode-shell correspondence to higher dimensions and symmetry classes, providing a unified topological framework for gapless modes in various topological phases, including semimetals and insulators.

Contribution

It generalizes the mode-shell correspondence to arbitrary dimensions and symmetry classes, linking phase space gapless modes to shell invariants without requiring translation invariance.

Findings

Derived expressions for mode counts depending on dispersion relation dimension.

Connected shell topology to specific topological invariants like Chern and winding numbers.

Validated the framework on models of semimetals and insulators across different dimensions.

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.