Dynamics and topology of non-Hermitian elastic lattices with non-local feedback control interactions

We investigate non-Hermitian elastic lattices characterized by non-local feedback control interactions. In one-dimensional lattices, we show that the proportional control interactions produce complex dispersion relations characterized by gain and loss in opposite propagation directions. Depending on the non-local nature of the control interactions, the resulting non-reciprocity occurs in multiple frequency bands characterized by opposite non-reciprocal behavior. The dispersion topology is also 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 dependent wave amplification. Moreover, the non-Hermitian skin effect manifests as modes localized at the boundaries of finite lattice strips, whose combined effect in two directions leads to the presence of bulk modes localized at the corners of finite two-dimensional lattices. Our results describe fundamental properties of non-Hermitian elastic lattices, and open new possibilities for the design of metamaterials with novel functionalities related to selective wave filtering, amplification and localization. The results also suggest that feedback interactions may be a useful strategy to investigate topological phases of non-Hermitian systems.


I. INTRODUCTION
systems are localized at a boundary, in sharp contrast with the extend Bloch modes of Hermitian counterparts. This intriguing feature of non-Hermitian lattices has recently been experimentally demonstrated using topoelectrical circuits [66] and quantum walks of single photons [67]. Further theoretical investigations have also shown higher order skin modes localized at corners and edges of 2D and 3D non-Hermitian lattices [68].
While most studies have so far focused on non-Hermitian optical and condensed matter systems, a few works have explored non-Hermiticity in elastic and acoustic media, most of which focus on PT phase transitions and exceptional points [42,[69][70][71][72][73][74][75]. More recently, feedback control has been pursued to establish non-reciprocal interactions in a mechanical metamaterial that emulates the non-Hermitian Su-Schrieffer-Heeger (SSH) model [43]. Such setting was used to experimentally demonstrate the existence of zero-frequency edge states in the non-Hermitian topological phase, and also to realize unidirectional wave amplification [44]. Motivated by these notable contributions, we here investigate a family of 1D and 2D elastic lattices with non-local, proportional feedback interactions and explore a series of unconventional phenomena stemming from their non-Hermiticity. Starting from a wave propagation perspective, we demonstrate that the frequency bands of 1D lattices are entirely non-reciprocal, due to the presence of gain and loss in opposite propagation directions. Such behavior is tunable based on the non-locality of the feedback interactions, which can be exploited to establish multiple frequency bands with interchanging non-reciprocal behavior.
We also show that the bulk eigenmodes of finite lattices are localized at a boundary according to the non-Hermitian skin effect, and that their localization edge is well predicted by the winding number of the complex dispersion bands, which is aligned with recent findings on quantum lattices [53]. Our analysis is then extended to 2D lattices where non-reciprocity manifests itself as a preferential direction for wave amplification, which is defined by the control interactions. We show that the non-local control in 2D lattices establishes multiple non-reciprocal frequency/wavenumber bands with different preferential directions of amplification. Finally, we investigate skin modes in finite lattice strips and show that their combined effect in two directions leads to bulk modes localized at the corners of finite 2D lattices. Our work provides fundamental perspectives on a new class of non-Hermitian elastic lattices with feedback interactions and contributes to recent efforts in exploring non-Hermiticiy for the design of metamaterials with novel functionalities [43,44]. This paper is organized as follows: following this introduction, the analysis of wave propagation and topological properties of 1D lattices with feedback interactions is presented.
Next, results are extended to 2D lattices where directional wave amplification and bulk corner modes are demonstrated. Finally, we summarize the main results of the work and outline future research directions.

TIONS
We consider 1D elastic lattices of equal masses m, separated by a unit distance, and connected by springs of equal stiffness k (Fig. 1). Control interactions are introduced by considering an additional force, applied to the n-th mass, that reacts proportionally to the elongation of a spring at location n − a (a ∈ I). This force is expressed as f n = k c (u n−a − u n−(a+1) ), where k c denotes the proportional control gain, and u n is the displacement of mass n along the x axis. In the absence of external forces, the equation governing the harmonic motion of mass n is given by The considered lattices are non-Hermitian since their dynamic stiffness matrix D = K − ω 2 M is real but not symmetric, i.e. D T = D. These lattices are non-conservative systems where gain and loss are introduced by the feedback interactions, leading to intriguing properties discussed throughout this paper. Although active components would be required for a practical implementation, these systems can be mathematically treated in a linear and autonomous form (as in Eqn. (1)), which motivates the investigations presented herein in terms of non-reciprocity and of topological properties of the bulk bands and their relation to the Non-Hermitian skin effect [62][63][64][65].

A. Dispersion relations and non-reciprocity
Wave propagation is investigated by imposing a Bloch-wave solution of the form u n = U e i(ωt−µn) , where ω and µ respectively denote angular frequency and non-dimensional wavenumber. Substitution in Eqn. (1) yields the dispersion relation where Ω = ω/ω 0 is a normalized frequency, with ω 0 = k/m, and γ c = k c /k. The feedback interaction makes the right-hand side of Eqn. (2) generally complex, which results in complex frequencies Ω = Ω r + iΩ i that come in pairs {Ω, −Ω}. Without loss of generality we focus on the solution Ω with positive real part (Ω r > 0), which corresponds to a wave u n = U e i(Ωrτ −µn) e −Ω i τ , (τ = tω 0 ), that travels along the positive (negative) x direction when µ is positive (negative), and that is exponentially attenuated (amplified) in time when Ω i is positive (negative).
We first investigate the case of local control (a = 0), i.e. with the feedback force proportional to the elongation of the left adjacent spring. (2) that Ω 2 r (µ) = Ω 2 r (−µ) and Ω 2 i (µ) = −Ω 2 i (−µ). By using basic properties related to the square roots of a complex number (not described here for brevity), one can confirm reciprocity for the real part of the dispersion (Ω r (µ) = Ω r (−µ)), and non-reciprocity for the imaginary part (Ω i (µ) = −Ω i (−µ)).
Due to this property, the amplification and attenuation wavenumber ranges defined by the imaginary part of the dispersion can be translated to the real frequency dispersion curves by matching the corresponding wavenumber intervals ( Fig. 3(a)). The procedure highlights two non-reciprocal frequency bands; the first amplifies waves traveling to the left, while the latter amplifies waves traveling to the right. In general, when considering higher a values the number of non-reciprocal bands increases, usually being equal to a + 1. For example, the dispersion for a = 3, γ c = 0.1 displayed in Fig. 3(c) exhibits a total of four non-reciprocal frequency bands, as highlighted by shaded green and pink regions.  Fig. 3(b)), and of two wave packets along each direction for a = 3 ( Fig. 3(d)). The corresponding 2D-FTs are superimposed to the dispersion curves in Figs. 3(a,c), confirming the predicted amplification bands. We also note that the amplification of the wave packets is intensified around wavenumbers associated with local minima of Ω i , corresponding to the largest time amplification exponents.
where Ω = ∂Ω/∂µ, and the base frequency Ω b is an arbitrary point in the complex plane not belonging to the dispersion band [53], i.e. Ω b = Ω(µ). Geometrically, the winding number where their eigenfrequencies lie. The modes inside the first region (represented by mode I in Fig. 6(b)) are localized at the interface, since in that region ν = 1 for sub-lattice A implies a tendency of localization towards its right, while ν − 1 for sub-lattice B implies a tendency for localization towards its left. Modes inside a second large region (represented by mode III in Fig. 6(b)) exhibit an opposite behavior: ν = −1 is associated with sub-lattice A, which implies a tendency for localization to its left, while ν = 1 is associated with sub-lattice B, which implies a tendency for localization to its right. The modes inside this region are therefore "double skin modes" simultaneously localized at both boundaries. In a small region between the two larger regions the modes are characterized by ν = 1 for both sublattices, and a slight tendendy of amplification towards the right boundary is observed (mode II).
A final set of modes represented by mode IV lie in a region outside the dispersion loop for sub-lattice A, and in a region with ν = 1 for sub-lattice B, which results in localization to the right.

TIONS
We now extend the study to 2D lattices consisting of equal masses m connected by springs k, separated by a unit distance in both x and y directions. Each mass moves along the perpendicular z direction (Fig. 7), so that the springs react with a force proportional to the relative vertical motion of neighboring masses.

A. Dispersion relations, non-reciprocity and and directionality
We impose Bloch wave solutions in Eqn. (4) of the form u n,m = U e i(ωt−µxn−µym) , where µ x and µ y are the wave vector components along x and y, respectively. This gives: where again Ω = ω/ω 0 , with ω 0 = k/m, while γ x = k cx /k and γ y = k cy /k. Similar to the one-dimensional case, we consider the solution with Ω r > 0 to represent the dispersion, such that Ω i < 0 is associated with wave amplification, while Ω i > 0 with attenuation.
For the local control case (a = 0), Figs. 8(a,b) display the real and imaginary iso-frequency contours of the dispersion surfaces of a lattice with γ x = γ y = 0.1. While the real part ( Fig. 8(a)) closely resembles that of a passive 2D lattice [2], the imaginary part of the frequency contours ( Fig. 8(b)) exhibits directional dependent attenuation and amplification zones. In particular, a region for which Ω i < 0 is identified in the third quadrant of the µ x , µ y plane ( Fig. 8(b)), revealing a range of directions of wave amplification. This is further illustrated by considering a frequency of Ω = 0.7, whose corresponding contour in the Ω r map, highlighted by the thick black line in Fig. 8(a), is approximately circular, possibly suggesting isotropic propagation. However, the wave vector components at this frequency (also highlighted by the thick black circle in Fig. 8 Similar to the 1D case, non-local feedback interactions in 2D lattices result in multiple non-reciprocal bands. This is illustrated for a lattice with a = 1 and γ x = γ y = 0.3 in Fig. 10. The real part of the dispersion displayed in Fig. 10(a) is similar to that of the local case ( Fig. 8(a)). In contrast, the imaginary component of the dispersion (Fig. 10(b)) exhibits different regions of amplification and attenuation when compared to the local case ( Fig. 8(b)). Contours at three different Ω r values are highlighted in Fig. 10

B. Bulk topology, skin modes and corner modes
We extend the winding number analysis conducted for 1D lattices to describe the topological properties of 2D lattices and demonstrate the presence of skin edge and corner modes.
We start by considering a finite lattice strip and show that its dispersion is associated with modes localized at one of the boundaries, that are either amplified or attenuated as they propagate along the other (infinite) direction. We then show that the combined effect of localization for finite strips in two directions (x and y) produce modes that are localized at the corners of finite lattices.
As a representative case, we consider the set of parameters a = 1, γ x = γ y = 0.3, corresponding to the lattice whose wave properties are described in Fig. 10. We consider a finite lattice strip with N = 30 masses along the x direction, and infinite along the y direction. To understand the topological properties and localization of the strip modes, we first consider a single wavenumber µ y = −π, for which the dispersion Ω(µ x , µ y = −π) is displayed in Fig. 11(a), while its projection on the complex plane defines a loop represented by red lines in Fig. 11(b). The eigenfrequencies of the finite strip for µ y = −π are represented by dots in Fig. 11(b), while a few representative modes marked by blue circles in Fig. 11(b) have their mode shapes displayed in Fig. 11(c), revealing localization at the boundaries.
Our analysis reveal that the localization of the strip modes for a given µ y is related to the topology of the dispersion Ω(µ x ) at that µ y value. In Fig. 11(b), blue and red zones again define regions for which ν = −1 and ν = 1, and similar to the 1D lattices, modes of the finite strip whose eigenfrequencies lie inside such regions are respectively localized at the left (blue dots) or right (red dots) boundary (Fig. 11(c)). to the behavior of the 1D lattice with a = 1 (Fig. 3(a)). This opens the possibility of establishing non-reciprocal wave amplification as demonstrated in Fig. 3   The behavior predicted by the dispersion analysis is illustrated by snapshots representing the lattice response to a broadband input (Fig. 3(e,f)) at subsequent time instants (f,g).