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This paper considers generalised network, intended as networks where (a) the edges connecting the nodes are nonlinear, and (b) stochastic processes are continuously indexed over both vertices and edges. Such topological structures are…

Recently, anomalous Floquet topological phases without static counterparts have been observed in different systems, where periodically driven models are realized to support a winding number of 1 and a pair of edge modes in each quasienergy…

Topological insulators are unique physical structures that are insulators in their bulk, but support currents at their edges which can be unidirectional and topologically protected from scattering on disorder and inhomogeneities. Photonic…

We consider two dimensional systems in which edge states coexist with a gapless bulk. Such systems may be constructed, for example, by coupling a gapped two dimensional state of matter that carries edge states to a gapless two dimensional…

Under the action of a weak constant force a wavepacket in periodic potential undergoes periodic oscillations in space, returning to the initial position after one oscillation cycle. This wave phenomenon, known as Bloch oscillations (BOs),…

We study the influence of topology on the quench dynamics of a system driven across a quantum critical point. We show how the appearance of certain edge states, which fully characterise the topology of the system, dramatically modifies the…

Synchronization of coupled oscillators is a ubiquitous phenomenon found throughout nature. Its robust realization is crucial to our understanding of various nonlinear systems, ranging from biological functions to electrical engineering. On…

Symmetries -- whether explicit, latent, or hidden -- are fundamental to understanding topological materials. This work introduces a prototypical spring-mass model that extends beyond established canonical models, revealing topological edge…

The presence of a topologically non-trivial discrete invariants implies the existence of gapless modes in finite samples, but it does not necessarily imply their localization. The disappearance of the indirect energy gap in the bulk…

The edges of a two-dimensional topological phase of matter serve as a platform underlying its low-energy dynamics. The topology of the bulk phase dictates the structure of the gapless modes. Proximitizing boundary modes to another boundary,…

Unlike the edge state of a topological insulator where its energy level lives in the bulk energy gap, the edge state of a topological semimetal hides in the bulk spectrum and is difficult to be identified by the energy. We investigate the…

Soft elastic filaments that can be stretched, bent and twisted exhibit a range of topologically and geometrically complex morphologies that include plectonemes, solenoids, knot-like and braid-like structures. We combine numerical…

We investigate topological edge states in one-dimensional off-diagonal mosaic lattices, where nearest-neighbor hopping amplitudes are modulated periodically with period $\kappa>1$. Analytically, we demonstrate that discrete edge states…

Many topological phenomena first proposed and observed in the context of electrons in solids have recently found counterparts in photonic and acoustic systems. In this work, we demonstrate that non-Abelian Berry phases can arise when…

Conventional topological insulators exhibit exotic gapless edge or surface states, as a result of non-trivial bulk topological properties. In periodically-driven systems the bulk-boundary correspondence is fundamentally modified and…

We develop a geometric framework to study the structure and function of complex networks. We assume that hyperbolic geometry underlies these networks, and we show that with this assumption, heterogeneous degree distributions and strong…

In this work, we give a precise mathematical description of a fully chiral gapless edge of a 2d topological order (without symmetry). We show that the observables on the 1+1D world sheet of such an edge consist of a family of topological…

The rigidity of elastic networks depends sensitively on their internal connectivity and the nature of the interactions between constituents. Particles interacting via central forces undergo a zero-temperature rigidity-percolation transition…

A different type of topological phase dubbed topological inverse Anderson insulators is proposed, which is characterized by the disorder-induced extended bulk states from the flat-band localization and topological edge states. Based on the…

Periodic networks on the verge of mechanical instability, called Maxwell lattices, are known to exhibit zero-frequency modes localized to their boundaries. Topologically polarized Maxwell lattices, in particular, focus these zero modes to…