Lyapunov exponents explain disorder-induced polarization and soliton teleportation in a mechanical Markov system

TL;DR

The paper introduces a space-to-time mapping that treats a highly disordered 1D mechanical lattice as a mechanical Markov system, with Lyapunov exponents controlling disorder-induced polarization of a zero mode (ZM) and enabling nonlinear soliton-like mobility. By constructing two solution branches $f_a,f_b$ (and analytic forms $f_\alpha,f_\beta$) and deriving a probabilistic recursion for rotor configurations, the authors predict exponential ZM localization toward a generating end and define stochastic protection against delocalization. They develop a discrete Markov-matrix approximation, perturbative analyses for low-frequency modes, and Hellmann–Feynman diagnostics to predict defect types and soliton transport, including chirality-dependent teleportation and defect-mediated spin-orbit coupling. These results provide a framework to design disorder-enabled linear and nonlinear responses in programmable metamaterials and suggest extensions to higher dimensions and non-Hermitian settings. Overall, Lyapunov exponents emerge as a compact, predictive language for encoding and engineering localization and dynamic transport in disordered mechanical systems.

Abstract

Using a mapping between spatial disorder and temporal stochasticity, we develop a new framework using Lyapunov exponents to explain exotic wave localization and mobility phenomena in disordered one-dimensional (1D) mechanical systems that can be constructed via a spatial analog of a Markov process, which we call ``mechanical Markov systems.'' We show that disorder induces robust polarization of zero modes (ZMs) in these mechanical Markov systems, and this phenomenon is explained using Lyapunov exponents. Remarkably, these ZMs become mobile solitons in the nonlinear regime despite the disorder-controlled localization of all other modes, and display a set of new nonlinear dynamics features including reflectionless chirality-dependent teleportation, which can also be explained using Lyapunov exponents. Our results establish the Markov formalism as a powerful tool to explain and design localization and dynamics in disordered mechanical systems, opening opportunities for programmable metamaterials with novel linear and nonlinear responses.

Figures (8)

• Figure 1: Mechanical Markov rotor chain construction and observations of ZM polarization and soliton dynamics. (A) A rotor chain system with geometric parameters ($l=1, r=1, a=0.5$) such that for any possible $\theta_n$, there will always be two possible solutions for $\theta_{n+1}$ (either $f_a, f_b$,condition: $a<l<r+a/2$). Coordinate labeling of this system is two-periodic. (B) The two possible solutions for $\theta_{n+1}$ as a function of $\theta_n$ (pink/thick, purple/thin) have four fixed points (red dots, star), at which ordered systems can be constructed (panel A constructed at the star, at which $\lambda(\star) = -0.570$). Disordered chains are constructed by choosing between the allowed solutions at random. Here, this is represented via a cobweb plot for a section (rotors 36 through 48) of the system in panel E. (C) The geometric mean of zero mode (ZM) amplitude ($\delta\theta_n$) in $10^3$ disordered ($p_a=0.5$) chains (black, solid, error bars $= 1\sigma$) has steady-state exponential decay that is in good agreement with theory decay rate (black, dashed). The left-localized Kane-Lubensky (KL) chain (red line), necessarily constructed at the one of the fixed points in (B), has a much faster decay rate (red, dashed) equal to the local Lyapunov exponent at those points (which are equal by symmetry, see SI Sec. \ref{['sym']}). (purple) The floppy mode amplitude of a particular rotor chain, shown in panel E. (D) The spatiotemporal map of soliton transport through this system shows defect regions of modified soliton transport, where an outgoing (incoming) soliton travels slowly (instantly) [I] or instantly (slowly) [II]. Colors show local rotor angular velocity $\dot{\theta}_{n}$ at time $t$. (E) The fifty rotor chain used in panels B-D, with solution choices ($a,b$) shown, and defect regions highlighted. The disorder-induced polarization is shown as black arrows, which points in the direction of increasing ZM amplitude, leading to quasi topological charges $[+]([-])$ where polarization converges (diverges).
• Figure 2: Stochastic protection of disorder-induced polarization in the 1D mechanical Markov rotor chain, and related design principles. (A) Disordered rotor chains ($p_a = p_b = 0.5$), in the large system limit, reach a steady-state rotor angle probability density, in which rotor angles where the solution functions have smaller slopes ($\sim$1-2 rads.) are significantly more probable than rotor angles with large slopes ($\sim$4-5 rads.). Analytic results for $\rho_{10}(\theta)$ (blue, from Eq. (\ref{['eq:theoryPDF']}) are in good agreement with simulation frequency (blue, mean of $10^5$ rotors) and the steady-state eigenvector a discrete Markov transition matrix of $10^4$ bins (green, from SI Sec. \ref{['disMarkov']}). Analytic results for $\rho_{2}(\theta),\rho_{3}(\theta)$ show rapid, convergence to the steady-state limit. Note that analytic results shown here are actually many discrete points for computational simplicity, but could, in principle, be calculated continuously. (B) Choosing with other branch probabilities (such as $p_a =0.6, p_b =0.4$) results in a different $\rho_{\text{s.s}}$. (C) Due to the skewed probability density (induced by $p_a=0.5$), rotor chains of length $n$ ($\theta_1$ is chosen from $\rho_{\text{s.s}}$ to avoid convergence effects) are exponentially unlikely to be more floppy at position $n$ than position $1$, as shown by mean probability ($\mu$) and standard error ($\sigma_\mu$) of $2*10^4$ simulations each of rotor chains of lengths $2,4,8,16,32$. (D) From the two possible solutions for $\theta_{n+1}$ ($f_a, f_b$), one can create two piecewise functions of $\theta_n$ (yellow, thick: $f_\alpha$, light blue: $f_\beta$), which result in deterministic successive solutions with positive and negative Lyapunov exponents respectively (within absorbing subsets $u, v$). By choosing repeated solutions from $f_{\alpha (\beta)}$, ZM amplitude is guaranteed to decrease (increase). By choosing subsequent angle solutions (colored arrows) initially from $f_\beta$, then from $f_\alpha$ (transition marked by black arrow), one can construct a chain whose ZM initial grows and then decays. (E, F) By switching back and forth between these solutions branches, one can create ZMs that are localized to the center (i) or both ends (ii) of a chain, or more generally, any pattern that is a combination of exponential growth/decay (iii, iv).
• Figure 3: Perturbation theory explains these existence of qualitatively low-frequency modes and their localizations. (A) The displacement modes (solid points, right singular vectors of $C$, the compatibility matrix) of the system in Fig. \ref{['fig:fig1']}E include the highly localized (high inverse participation ratio, I.P.R., $\sum_i x_i^4/\sum_i x_i^2$) topological zero-frequency mode (purple), as well as two relatively localized, qualitatively low-frequency modes (green, blue). First order perturbation theory results (dashed open circles) reasonably approximate these low-frequency modes. All modes are more localized than the usual I.P.R. of $~1/N$ of plane waves (horizontal dashed line). (B) The stress modes (solid dots, left singular vectors of $C$) include no topological states of self stress (no $\omega=0$, as expected by Maxwell counting), but do include two localized, low-frequency modes (orange, red). These frequencies are the same as low-frequency deformation modes, as the frequency is their shared singular value of $C$. (C) The right (left) singular vectors of $C$ that represent low-frequency displacement (stress) modes exponentially localize to local maxima (minima) of the topological ZM, all with roughly equal decay rates. (D) First order perturbation theory reproduces these localizations for both low-frequency displacement (circle markers) and stress (square markers) modes. (E) For displacement modes, perturbation theory is applied by removing edges 20 and 38, which are at ZM minima (orange, dashed) to obtain a zeroth order system. These removed edges are reintroduced as a perturbation. (F) The zeroth order system has three degenerate, non-overlapping ZMs ($\ket{\phi_j^{}}$, eigenstates of zeroth order Hamiltonian $D_0$), dashed lines indicate where edges were removed.
• Figure 4: Regular soliton transport in disordered rotor chains is interrupted at "defects," where soliton transport is either slow or an instant teleport. (A) Transport of a counterclockwise soliton (pink hue $\propto |\dot{\theta}|$, same as Fig. \ref{['fig:fig1']}A) with and overlay of locations of the quasi topological charges (rotor with highest amplitude of corresponding singular vector). Each "defect" region is comprised of a positive ([$+$]) and negative ([$-$]) quasi topological charge, which here are listed with the subscript of the defect they correspond to. (B) The frequencies of quasi ZMs and quasi SSSs from (A), note that coupled charges of the same defect have the exact same frequency due to sharing the same singular value of $C$. During "slow transport" (shaded boxes), quasi topological charges annihilate, raising the frequency of these usually "low" frequency modes to be on par with other "high" frequency modes (next four lowest shown in black). (C/D) The behavior of a defect (slow or instant soliton transport) is determined by the domain of angles swept out by the leftmost angle ($\theta_{16}$ for I and $\theta_{31}$ for II) during transport. If the leftmost angle traverses through a rapid cycle (narrow shaded region of angles in the leftmost rotor in which a middle rotor completes nearly a complete rotation), the defect will have slow transport (C). If the leftmost angle does not traverse through a rapid cycle, the defect will have instant transport (D). (E) Transport of a clockwise soliton on the exact same system. Here, the behavior of defects is opposite from the counterclockwise case in panel A (slow $\leftrightarrow$ instant) because the compliment set of angles is swept out by $\theta_{16}, \theta_{38}$, negating the inclusion of a rapid cycle. (F/G) Zoomed in panels of the dashed boxes in A/E with shaded box to show mode annihilation.
• Figure 5: Diagram of angles and vectors used in the derivation of SSS Lyapunov exponents.