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Abstract We investigate the spectral properties of one-dimensional spatially modulated nonlinear phononic lattices, and their evolution as a function of amplitude. In the linear regime, the stiffness modulations define a family of periodic and quasiperiodic lattices whose bandgaps host topological edge states localized at the boundaries of finite domains. With cubic nonlinearities, we show that edge states whose eigenvalue branch remains within the gap as amplitude increases remain localized, and therefore appear to be robust with respect to amplitude. In contrast, edge states whose corresponding branch approaches the bulk bands experience de-localization transitions. These transitions are predicted through continuation studies on the linear eigenmodes as a function of amplitude, and are confirmed by direct time domain simulations on finite lattices. Through our predictions, we also observe a series of amplitude-induced localization transitions as the bulk modes detach from the nonlinear bulk bands and become discrete breathers that are localized in one or more regions of the domain. Remarkably, the predicted transitions are independent of the size of the finite lattice, and exist for both periodic and quasiperiodic lattices. These results highlight the co-existence of topological edge states and discrete breathers in nonlinear modulated lattices. Their interplay may be exploited for amplitude-induced eigenstate transitions, for the assessment of the robustness of localized states, and as a strategy to induce discrete breathers through amplitude tuning. 

Abstract We investigate the dynamic behavior of lattices with disorder introduced through non-local network connections. Inspired by the Watts–Strogatz small-world model, we employ a single parameter to determine the probability of local connections being re-wired, and to induce transitions between regular and disordered lattices. These connections are added as non-local springs to underlying periodic one-dimensional (1D) and two-dimensional (2D) square, triangular and hexagonal lattices. Eigenmode computations illustrate the emergence of spectral gaps in various representative lattices for increasing degrees of disorder. These gaps manifest themselves as frequency ranges where the modal density goes to zero, or that are populated only by localized modes. In both cases, we observe low transmission levels of vibrations across the lattice. Overall, we find that these gaps are more pronounced for lattice topologies with lower connectivity, such as the 1D lattice or the 2D hexagonal lattice. We then illustrate that the disordered lattices undergo transitions from ballistic to super-diffusive or diffusive transport for increasing levels of disorder. These properties, illustrated through numerical simulations, unveil the potential for disorder in the form of non-local connections to enable additional functionalities for metamaterials. These include the occurrence of disorder-induced spectral gaps, which is relevant to frequency filtering devices, as well as the possibility to induce diffusive-type transport which does not occur in regular periodic materials, and that may find applications in dynamic stress mitigation. 

Abstract Twisted bilayered systems such as bilayered graphene exhibit remarkable properties such as superconductivity at magic angles and topological insulating phases. For generic twist angles, the bilayers are truly quasiperiodic, a fact that is often overlooked and that has consequences which are largely unexplored. Herein, we uncover that twistedn-layers host intrinsic higher dimensional topological phases, and that those characterized by second Chern numbers can be found in twisted bi-layers. We employ phononic lattices with interactions modulated by a second twisted lattice and reveal Hofstadter-like spectral butterflies in terms of the twist angle, which acts as a pseudo magnetic field. The phason provided by the sliding of the layers lives on 2n-tori and can be used to access and manipulate the edge states. Our work demonstrates how multi-layered systems are virtual laboratories for studying the physics of higher dimensional quantum Hall effect, and can be employed to engineer topological pumps via simple twisting and sliding. 

Abstract We investigate non-Hermitian elastic lattices characterized by non-local feedback interactions. In one-dimensional lattices, proportional feedback produces non-reciprocity associated with complex dispersion relations characterized by gain and loss in opposite propagation directions. For non-local controls, such non-reciprocity occurs over multiple frequency bands characterized by opposite non-reciprocal behavior. The dispersion topology is investigated with focus on winding numbers and non-Hermitian skin effect, which manifests itself through bulk modes localized at the boundaries of finite lattices. In two-dimensional lattices, non-reciprocity is associated with directional wave amplification. Moreover, the combination of skin effect in two directions produces modes that are localized at the corners of finite two-dimensional lattices. Our results describe fundamental properties of non-Hermitian elastic lattices, and suggest new possibilities for the design of meta materials with novel functionalities related to selective wave filtering, amplification and localization. The considered non-local lattices also provide a platform for the investigation of topological phases of non-Hermitian systems.