TY  - JOUR
PB  - SciPost Foundation
DO  - 10.21468/SciPostPhys.18.6.193
TI  - Mode-Shell correspondence, a unifying phase space theory in topological physics - Part II: Higher-dimensional spectral invariants
PY  - 2025/06/18
UR  - https://www.scipost.org/SciPostPhys.18.6.193
JF  - SciPost Physics
JA  - SciPost Phys.
VL  - 18
IS  - 6
SP  - 193
A1  - Jezequel, Lucien
AU  - Delplace, Pierre
AB  - The mode-shell correspondence relates the number $\mathcal{I}_M$ of gapless modes in phase space to a topological \textit{shell invariant} $\mathcal{I}_S$ defined on a closed surface - the shell - surrounding those modes, namely $\mathcal{I}_M=\mathcal{I}_S$. In part I, we introduced the mode-shell correspondence for zero-modes of chiral symmetric Hamiltonians (class AIII). In this part II, we broaden 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, $2D$ massless Dirac and $3D$-Weyl cones, within the same formalism. We provide an expression for $\mathcal{I}_M$ 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.
ER  -

@Article{10.21468/SciPostPhys.18.6.193,
	title={{Mode-Shell correspondence, a unifying phase space theory in topological physics - Part II: Higher-dimensional spectral invariants}},
	author={Lucien Jezequel and Pierre Delplace},
	journal={SciPost Phys.},
	volume={18},
	pages={193},
	year={2025},
	publisher={SciPost},
	doi={10.21468/SciPostPhys.18.6.193},
	url={https://scipost.org/10.21468/SciPostPhys.18.6.193},
}