Contributor info: Dr Pierre Delplace

Thibaud Louvet, Pierre Delplace, Mark-Oliver Goerbig, David Carpentier

SciPost Phys. 15, 129 (2023) · published 3 October 2023 | Toggle abstract · pdf

Linear crossings of energy bands occur in a wide variety of materials. In this paper we address the question of the quantization of the Berry winding characterizing the topology of these crossings in dimension $D=2$. Based on the historical example of $2$-bands crossing occuring in graphene, we propose to relate these Berry windings to the topological Chern number within a $D=3$ dimensional extension of these crossings. This dimensional embedding is obtained through a choice of a gap-opening potential. We show that the presence of an (emergent) $\mathcal{PT}$ symmetry, local in momentum and antiunitary, allows the quantization of the Berry windings as multiples of $\pi$. We illustrate this quantization mechanism on a variety of three-band crossings.

Pierre Delplace

SciPost Phys. Lect. Notes 39 (2022) · published 10 March 2022 | Toggle abstract · pdf

This lecture note adresses the correspondence between spectral flows, often associated to unidirectional modes, and Chern numbers associated to degeneracy points. The notions of topological indices (Chern numbers, analytical indices) are introduced for non specialists with a wave physics or condensed matter background. The correspondence is detailed with several examples, including the Dirac equations in two dimensions, Weyl fermions in three dimensions, the shallow water model in fluid dynamics and other generalizations.

Pierre A. L. Delplace

SciPost Phys. 8, 081 (2020) · published 29 May 2020 | Toggle abstract · pdf

Using elementary graph theory, we show the existence of interface chiral modes in random oriented scattering networks and discuss their topological nature. For particular regular networks (e.g. L-lattice, Kagome and triangular networks), an explicit mapping with time-periodically driven (Floquet) tight-binding models is found. In that case, the interface chiral modes are identified as the celebrated anomalous edge states of Floquet topological insulators and their existence is enforced by a symmetry imposed by the associated network. This work thus generalizes these anomalous chiral states beyond Floquet systems, to a class of discrete-time dynamical systems where a periodic driving in time is not required.