Contributor info: Mr Lucien Jezequel

Lucien Jezequel, Pierre Delplace

SciPost Phys. 18, 193 (2025) · published 18 June 2025 | Toggle abstract · pdf

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.

Xinxin Guo, Lucien Jezequel, Mathieu Padlewski, Hervé Lissek, Pierre Delplace, Romain Fleury

SciPost Phys. 18, 034 (2025) · published 27 January 2025 | Toggle abstract · pdf

Nonlinear topology has been much less inquired compared to its linear counterpart. Existing advances have focused on nonlinearities of limited magnitudes and fairly homogeneous types. As such, the realizations have rarely been concerned with the requirements for nonlinearity. Here we explore nonlinear topological protection by determining nonlinear rules and demonstrate their relevance in real-world experiments. We take advantage of chiral symmetry and identify the condition for its continuation in general nonlinear environments. Applying it to one-dimensional topological lattices, we show possible evolution paths for zero-energy edge states that preserve topologically nontrivial phases regardless of the specifics of the chiral nonlinearities. Based on an acoustic prototype design with non-local nonlinearities, we theoretically, numerically, and experimentally implement the nonlinear topological edge states that persist in all nonlinear degrees and directions without any frequency shift. Our findings unveil a broad family of nonlinearities compatible with topological non-triviality, establishing a solid ground for future drilling in the emergent field of nonlinear topology.

Lucien Jezequel, Pierre Delplace

SciPost Phys. 17, 060 (2024) · published 23 August 2024 | Toggle abstract · pdf

We propose a theory, that we call the mode-shell correspondence, which relates the topological zero-modes localised in phase space to a 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-dimensional 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.