Non-Hermitian topology in static mechanical metamaterials

Topological elastic waves provide novel and robust ways for manipulating mechanical energy transfer and information transmission, with potential applications in vibration control, analog computation, and more. Recently discovered higher-order topological insulators (HOTIs) with multidimensional and hierarchical edge states can further expand the capabilities of topological elastic waves. However, the effects of nonlinearity on elastic HOTIs remain elusive. In this paper, we propose a nonlinear elastic higher-order topological Kagome lattice. After briefly reviewing its linear properties, we explore the effects of nonlinearity on the higher-order band topology and topological states. To do this, we have developed a method to calculate approximate nonlinear modes in order to identify the bulk polarization and probe the higher-order topological phase in the nonlinear lattice. We find that nonlinearity induces unusual delocalization of topological corner states, band crossing, and higher-order topological phase transition. The delocalization reveals that intracell hardening nonlinearity leads to direct delocalization of topological corner states while intracell softening nonlinearity first enhances and then reduces localization. The nonlinear higher-order topological phase is amplitude dependent, and we demonstrate a transition from a trivial to a non-trivial phase, enabling amplitude induced topological corner and edge states. Additionally, this phase transition corresponds to the closing and reopening of the bandgap, accompanied by an unusual band crossing. By examining the band topology before and after the band crossing, we find that the bulk polarization becomes quantized with respect to amplitude and can predict higher-order topological phases in nonlinear lattices. The obtained results are expected to be beneficial for the development of tunable and robust elastic wave devices.

Section: Nonlinear Higher-order Topological Statesmentioning

confidence: 99%

“…This unique phenomenon can be dated back to the discovery of the quantum Hall effect, which opened up a new paradigm in condensed matter physics [2] beyond electronic systems, e.g. acoustical [3], photonic [4], and mechanical systems [5][6][7]. Traditionally, a n-dimensional (nD) TI only supports (n-1)D edge states [1].…”

Section: Introductionmentioning

confidence: 99%

Delocalization and higher-order topology in a nonlinear elastic lattice

Yi,

Chen

2024

“…For instance, in photonic lattices, non-reciprocal hopping is often controlled using amplitude and phase modulators between lattice sites [29]. Acoustic systems achieve non-reciprocal hopping through directional amplifiers [74], while in mechanical metamaterials, modulation of bar stiffness facilitates the engineering of non-reciprocity, as observed in studies on non-Hermitian topology in static mechanical metamaterials [79].…”

Section: Appendix F Derivation Of the Paired Winding Numbermentioning

confidence: 99%

Emergence of two-fold non-Hermitian spectral topology through synthetic spin engineering

Sarkar,

Banerjee,

Narayan

2024

The union of topology and non-Hermiticity has led to the unveiling of many intriguing phenomena. We introduce a synthetic spin-engineered model belonging to symmetry class AI, which is a rare occurrence, and demonstrate the emergence of a multi-fold spectral topology. As an example of our proposal, we engineer non-Hermiticity in the paradigmatic Su–Schrieffer–Heeger (SSH) model by introducing a generalized synthetic spin, leading to an emergent two-fold spectral topology that governs the decoupled behaviour of the corresponding non-Hermitian skin effect. As a consequence of the spin choice, our model exhibits a rich phase diagram consisting of distinct topological phases, which we characterize by introducing the notion of paired winding numbers, which, in turn, predict the direction of skin localization under open boundaries. We demonstrate that the choice of spin parameters enables control over the directionality of the skin effect, allowing for it to be unilateral or bilateral. Furthermore, we discover non-dispersive flat bands emerging within the inherent SSH model framework, arising from the spin-engineering approach. We also introduce a simplified toy model to capture the underlying physics of the emergent flat bands and direction-selective skin effect. As an illustration of experimental feasibility, we present a topoelectric circuit that faithfully emulates the underlying spin-engineered Hamiltonian, providing a viable platform for realizing our predicted effects. Our findings pave way for the exploration of unconventional spectral topology through spin-designed models.

“…This finding suggests that the (Schrödinger) static mode is very much similar as the (Lotka-Volterra) static mode, because these two zero-frequency solutions are obtained by substituting i∂ t Ψ (i) r = 0 and ∂ t Ψ (i) r = 0 in the Schrödinger and Lotka-Volterra models, respectively. The notion of drawing a comparison between the static solution of the Lotka-Volterra and Schrödinger models originates from seminal works [4][5][6][7][8][9][10][11][109][110][111][112], which established an analogy between the static properties of the elastic compatibility matrix and the static solutions in chiral-symmetric Schrödinger equations. The topological properties of the elastic compatibility matrix are defined by introducing an auxiliary chiral-symmetric Schrödinger equation, where we can compute the topological index of this auxiliary Schrödinger equation.…”

Section: Applications To the Static Modes In Other Nonlinear Modelsmentioning

confidence: 99%

Topological boundary modes in nonlinear dynamics with chiral symmetry

Zhou

2024

Particle-hole symmetry and chiral symmetry play a pivotal role in multiple areas of physics, yet they remain unstudied in systems with nonlinear interactions whose nonlinear normal modes do not exhibit U ( 1 ) -gauge symmetry. In this work, we establish particle-hole symmetry and chiral symmetry in such systems. Chiral symmetry ensures the quantization of the Berry phase of nonlinear normal modes and categorizes the topological phases of nonlinear dynamics. We show topologically protected static boundary modes in chiral-symmetric nonlinear systems. Furthermore, we demonstrate amplitude-induced nonlinear topological phase transition in chiral-symmetric nonlinear dynamics. Our theoretical framework extends particle-hole and chiral symmetries to nonlinear dynamics, whose nonlinear modes do not necessarily yield U ( 1 ) -gauge symmetry.