arXiv subjects

Pierre Delplace

Publications and source records attributed to Pierre Delplace.

At least 37 records · Page 2Linked to original sources

arXiv2022

How do CPT-like symmetries shape the stability of geophysical flows?

We examine the role discrete symmetries, time-reversal and mirror symmetries, play in the context of geophysical waves and instabilities. By looking at three special cases from the two-layer quasi-geostrophic model as well as developing a general framework for translating real-space transformations to Fourier-space we are able to: 1) show that baroclinic instability is an example of spontaneous parity-time symmetry breaking; 2) show that pure parity symmetry for a fluid system is exactly analogous to charge-conjugation-parity symmetry in a condensed matter system; and 3) show that when a pure parity symmetry is broken, this is associated with the suppression of wave propagation. Further, in the latter case, instability can arise without a corresponding symmetry breaking. This study highlights the role of symmetry breaking behind the dynamics of geophysical waves and instabilities.

physics.ao-ph↗

arXiv2022

Nonlinear edge modes from topological 1D lattices

We propose a method to address the existence of topological edge modes in one-dimensional (1D) nonlinearlattices, by deforming the edge modes of linearized models into solutions of the fully nonlinear system. Forlarge enough nonlinearites, the energy of the modified edge modes may eventually shift out of the gap, leadingto their disappearance. We identify a class of nonlinearities satisfying a generalised chiral symmetry wherethis mechanism is forbidden, and the nonlinear edge states are protected by a topological order parameter.Different behaviours of the edge modes are then found and explained by the interplay between the nature of thenonlinarities and the topology of the linearized models.

nlin.PS↗

arXiv2021

Quantized Berry winding from an emergent $\mathcal{PT}$ symmetry

Linear crossing 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 of 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 us to relate $D=3$ Chern numbers to $D=2$ Berry windings quantized as multiple of $π$. We illustrate this quantization mechanism on a variety of three-band crossings.

cond-mat.mes-hall↗

arXiv2021

Symmetry-protected multifold exceptional points and their topological characterization

We investigate the existence of higher order exceptional points (EPs) in non-Hermitian systems, and show that $μ$-fold EPs are stable in $μ-1$ dimensions in the presence of anti-unitary symmetries that are local in parameter space, such as e.g. PT or CP symmetries. This implies in particular that 3-fold and 4-fold symmetry-protected EPs are stable respectively in 2 and 3 dimensions. The stability of such exceptional points is expressed in terms of the homotopy properties of a "resultant vector" that we introduce. Our framework also allows us to rephrase the previously proposed $\mathbb{Z}_2$ index of PT and CP symmetric gapped phases beyond the realm of two-band models. We apply this general formalism to a frictional shallow water model that is found to exhibit 3-fold exceptional points associated with topological numbers $\pm1$. For this model, we also show different non-Hermitian topological transitions associated with these exceptional points, such as their merging and a transition to a regime where propagation becomes forbidden.

cond-mat.mes-hall↗

arXiv2021

Berry-Chern monopoles and spectral flows

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 and other generalizations.

cond-mat.mes-hall↗

arXiv2021

Wave topology brought to the coast

Since the pioneering work of Kelvin on Laplace tidal equations, a zoology of trapped waves have been found in the context of coastal dynamics. Among them, the one originally computed by Kelvin plays a particular role, as it is an unidirectional mode filling a frequency gap between different wave bands. The existence of such Kelvin waves is robust to changes in the boundary shape and in changes of the underlying model for the coast. This suggests a topological interpretation that has yet up to now remained elusive. Here we rectify the situation, by taking advantage of a reformulation of the shallow water dynamics that highlights an analogy with the celebrated Haldane model in condensed matter physics. For any profile of bottom topography, the number of modes that transit from one wave band to another in the dispersion relation is predicted by computing a first Chern number describing the topology of complex eigenmodes in a dual, simpler wave problem.

physics.flu-dyn↗

arXiv2021

Manifestation of Berry curvature in geophysical ray tracing

Geometrical phases, such as the Berry phase, have proven to be powerful concepts to understand numerous physical phenomena, from the precession of the Foucault pendulum to the quantum Hall effect and the existence of topological insulators. The Berry phase is generated by a quantity named Berry curvature, describing the local geometry of wave polarization relations and known to appear in the equations of motion of multi-component wave packets. Such a geometrical contribution in ray propagation of vectorial fields has been observed in condensed matter, optics and cold atoms physics. Here, we use a variational method with a vectorial Wentzel-Kramers-Brillouin (WKB) ansatz to derive ray tracing equations in geophysical waves and reveal the contribution of Berry curvature. We detail the case of shallow water wave packets and propose a new interpretation to the equatorial oscillation and the bending of rays in mid-latitude area. Our result shows a mismatch with the textbook scalar approach for ray tracing, by predicting a larger eastward velocity for Poincaré wave packets. This work enlightens the role of wave polarization's geometry in various geophysical and astrophysical fluid waves, beyond the shallow water model.

physics.flu-dyn↗

arXiv2021

Anomalous nonreciprocal topological networks: stronger than Chern insulators

Robustness against disorder and defects is a pivotal advantage of topological systems, manifested by absence of electronic backscattering in the quantum Hall and spin-Hall effects, and unidirectional waveguiding in their classical analogs. Two-dimensional (2D) topological insulators, in particular, provide unprecedented opportunities in a variety of fields due to their compact planar geometries compatible with the fabrication technologies used in modern electronics and photonics. Among all 2D topological phases, Chern insulators are to date the most reliable designs due to the genuine backscattering immunity of their non-reciprocal edge modes, brought via time-reversal symmetry breaking. Yet, such resistance to fabrication tolerances is limited to fluctuations of the same order of magnitude as their band gap, limiting their resilience to small perturbations only. Here, we tackle this vexing problem by introducing the concept of anomalous non-reciprocal topological networks, that survive disorder levels with strengths arbitrarily larger than their bandgap. We explore the general conditions to obtain such unusual effect in systems made of unitary three-port scattering matrices connected by phase links, and establish the superior robustness of the anomalous edge modes over the Chern ones to phase link disorder of arbitrarily large values. We confirm experimentally the exceptional resilience of the anomalous phase, and demonstrate its operation by building an ideal anomalous topological circulator despite its arbitrary shape and large number of ports. Our results pave the way to efficient, arbitrary planar energy transport on 2D substrates for wave devices with full protection against large fabrication flaws or imperfections.

cond-mat.mes-hall↗

arXiv2020

From the geometry of Foucault pendulum to the topology of planetary waves

The physics of topological insulators makes it possible to understand and predict the existence of unidirectional waves trapped along an edge or an interface. In this review, we describe how these ideas can be adapted to geophysical and astrophysical waves. We deal in particular with the case of planetary equatorial waves, which highlights the key interplay between rotation and sphericity of the planet, to explain the emergence of waves which propagate their energy only towards the East. These minimal ingredients are precisely those put forward in the geometric interpretation of the Foucault pendulum. We discuss this classic example of mechanics to introduce the concepts of holonomy and vector bundle which we then use to calculate the topological properties of equatorial shallow water waves.

physics.geo-ph↗

arXiv2020

Topological chiral modes in random scattering networks

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.

cond-mat.mes-hall↗

arXiv2020

Chiral Maxwell waves in continuous media from Berry monopoles

We propose a method to predict the existence of topologically protected electromagnetic chiral modes between continuous media described by Maxwell's equations. The number and character of these modes is related to topological charges (Berry monopoles) in parameter space. Unlike the approaches proposed so far, our description does not require a regularization parameter at large $\mathbf{k}$ nor the materials to be topological on their own. The predictions of the theory are confirmed by additional numerical simulations of interfaces of gyrotropic media.

physics.optics↗

arXiv2019

Topological transition in stratified fluids

Lamb waves are trapped acoustic-gravity waves that propagate energy over great distances along a solid boundary in density stratified, compressible fluids. They constitute useful indicators of explosions in planetary atmospheres. When the density stratification exceeds a threshold, or when the impermeability condition at the boundary is relaxed, atmospheric Lamb waves suddenly disappear. Here we use topological arguments to predict the possible existence of new trapped Lamb-like waves in the absence of a solid boundary, depending on the stratification profile. The topological origin of the Lamb-like waves is emphasized by relating their existence to two-band crossing points carrying opposite Chern numbers. The existence of these band crossings coincides with a restoration of the vertical mirror symmetry that is in general broken by gravity. From this perspective, Lamb-like waves also bear strong similarities with boundary modes encountered in quantum valley Hall effect. Our study shows that the presence of Lamb-like waves encode essential information on the underlying stratification profile in astrophysical and geophysical flows, which is often poorly constrained by observations.

physics.flu-dyn↗

arXiv2019

A bulk-interface correspondence for equatorial waves

Topology is bringing new tools for the study of fluid waves. The existence of unidirectional Yanai and Kelvin equatorial waves has been related to a topological invariant, the Chern number, that describes the winding of $f$-plane shallow water eigenmodes around band crossing points in parameter space. In this previous study, the topological invariant was a property of the interface between two hemispheres. Here we ask whether a topological index can be assigned to each hemisphere. We show that this can be done if the shallow water model in $f$-plane geometry is regularized by an additional odd-viscous term. We then compute the spectrum of a shallow water model with a sharp equator separating two flat hemispheres, and recover the Kelvin and Yanai waves as two exponentially trapped waves along the equator, with all the other modes delocalized into the bulk. This model provides an exactly solvable example of bulk-interface correspondence in a flow with a sharp interface, and offers a topological interpretation for some of the transition modes described by [Iga, Journal of Fluid Mechanics 1995]. It also paves the way towards a topological interpretation of coastal Kelvin waves along a boundary, and more generally, to an understanding of bulk-boundary correspondence in continuous media.

physics.flu-dyn↗

arXiv2019

Anomalous bulk-edge correspondence in continuous media

Topology plays an increasing role in physics beyond the realm of topological insulators in condensed mater. From geophysical fluids to active matter, acoustics or photonics, a growing family of systems presents topologically protected chiral edge modes. The number of such modes should coincide with the bulk topological invariant (e.g. Chern number) defined for a sample without boundary, in agreement with the bulk-edge correspondence. However this is not always the case when dealing with continuous media where there is no small scale cut-off. The number of edge modes actually depends on the boundary condition, even when the bulk is properly regularized, showing an apparent paradox where the bulk-edge correspondence is violated. In this paper we solve this paradox by showing that the anomaly is due to {ghost} edge modes hidden in the asymptotic part of the spectrum. We provide a general formalism based on scattering theory to detect all edge modes properly, so that the bulk-edge correspondence is restored. We illustrate this approach through the odd-viscous shallow-water model and the massive Dirac Hamiltonian, and discuss the physical consequences.

cond-mat.mes-hall↗

arXiv2017

Topological Origin of Equatorial Waves

Topology sheds new light on the emergence of unidirectional edge waves in a variety of physical systems, from condensed matter to artificial lattices. Waves observed in geophysical flows are also robust to perturbations, which suggests a role for topology. We show a topological origin for two celebrated equatorially trapped waves known as Kelvin and Yanai modes, due to the Earth's rotation that breaks time-reversal symmetry. The non-trivial structure of the bulk Poincaré wave modes encoded through the first Chern number of value $2$ guarantees existence for these waves. This invariant demonstrates that ocean and atmospheric waves share fundamental properties with topological insulators, and that topology plays an unexpected role in the Earth climate system.

cond-mat.mes-hall↗

arXiv2017

Phase rotation symmetry and the topology of oriented scattering networks

We investigate the topological properties of dynamical states evolving on periodic oriented graphs. This evolution, that encodes the scattering processes occurring at the nodes of the graph, is described by a single-step global operator, in the spirit of the Ho-Chalker model. When the successive scattering events follow a cyclic sequence, the corresponding scattering network can be equivalently described by a discrete time-periodic unitary evolution, in line with Floquet systems. Such systems may present anomalous topological phases where all the first Chern numbers are vanishing, but where protected edge states appear in a finite geometry. To investigate the origin of such anomalous phases, we introduce the phase rotation symmetry, a generalization of usual symmetries which only occurs in unitary systems (as opposed to Hamiltonian systems). Equipped with this new tool, we explore a possible explanation of the pervasiveness of anomalous phases in scattering network models, and we define bulk topological invariants suited to both equivalent descriptions of the network model, which fully capture the topology of the system. We finally show that the two invariants coincide, again through a phase rotation symmetry arising from the particular structure of the network model.

cond-mat.mes-hall↗

arXiv2017

Non-diffracting states in one-dimensional Floquet photonic topological insulators

One dimensional laser-written modulated photonic lattices are known to be particularly suitable for diffraction management purposes. Here, we address the connection between discrete non-diffracting states and topological properties in such devices through the experimental observation and identification of three classes of non-diffracting state. The first one corresponds to topologically protected edge states, recently predicted in Floquet topological insulators, while the second and third are both bulk modes. One of them testifies of a topological transition, although presenting topological features different from those of the edge states, whether the other one result from specific band structure engineering.

cond-mat.mes-hall↗

arXiv2015

Topological edge states in two-gap unitary systems: A transfer matrix approach

We construct and investigate a family of two-band unitary systems living on a cylinder geometry and presenting localized edge states. Using the transfer matrix formalism, we solve and investigate in details such states in the thermodynamic limit. Analitycity considerations then suggest the construction of a family of Riemman surfaces associated to the band structure of the system. In this picture, the corresponding edge states naturally wind around non contractile loops, defining by the way a topological invariant associated to each gap of the system.

cond-mat.mes-hall↗