Nonlinearity-induced transition in heat conduction through a topological metamaterial of rotors

TL;DR

This work analyzes heat conduction in a one-dimensional chain of rotors that exhibits topological mechanics in its linear form (the Kane–Lubensky chain) and nonlinear rotor dynamics. Using Langevin baths and Green's function methods for the linear model, the authors show that the gapped topological phases behave as thermal insulators with exponentially suppressed heat current, while the gapless transition yields ballistic conduction. In the nonlinear regime, the chain supports topological solitary waves and displays diffusive heat transport with finite conductivity across several parameter regimes, indicating that nonlinearity obscures the linear topological phase distinctions in classical transport. Overall, the study reveals a nonlinearity-induced transition from insulating to normal transport in topological rotor metamaterials and points to future avenues in quantum transport and phase detection beyond linear models.

Abstract

We investigate heat conduction in a one-dimensional chain of rigid rotors. The rotors are constrained to rotate in a plane about fixed pivot points and coupled by springs, such that in equilibrium, the neighboring rotors lie on opposite sides of the chain axis. The linearized limit of this model valid for small angular displacements, was first introduced by Kane and Lubensky (KL) as a topological mechanical insulator hosting zero-energy vibrational edge modes. We show that the linearized KL chain behaves as a thermal insulator at low temperature in both the topological phases with a finite band gap, and the heat current falls exponentially with the chain length. When the gap vanishes at the topological phase transition, the KL chain becomes a good thermal conductor and conducts heat ballistically. The chain of rotors for arbitrary angular displacements hosts nonlinear solitary waves and distinct topological mechanical phases. Our numerical analysis shows normal (diffusive) heat conduction in all topological phases of the nonlinear chain. Nevertheless, a finite thermal conductivity is achieved for different system sizes in different topological phases of this nonlinear chain.

Figures (3)

• Figure 1: A cartoon of the topological metamaterial of rotors. The chain is in an equilibrium position, where all rotors have a length $r=0.8$, and their pivot points are separated by a distance $a=1$. Out of the four rotors, the rotors at positions $x=0,2$ (blue) make an angle $\bar{\theta} = \pi/3$ with the $+y$ axis and the rotors at positions $x=1,3$ (green) make angle $\pi - \bar{\theta} = 2\pi/3$ with the $+y$ axis. The dotted box represents a unit-cell of the lattice. Since this is a right-leaning configuration, we may able to produce nonlinear solitary waves in this chain by perturbing the right-most rotor ($x=3$).
• Figure 2: System-size $(N)$ scaling of $JN$ in a nonlinear metamaterial of rotors for different values of equilibrium angle $\bar{\theta}$ of the rotors and $r/a$, where $J$ is steady-state heat current. Other parameters are $m=k=1, \gamma_L=\gamma_R=0.5, T_L=0.01,T_R=0.5$. $J$ shows diffusive scaling $(J \sim N^{-1})$ at longer $N$ for $\bar{\theta}=0, 0.97, \pi/2$ when $r/a=2$ and $\bar{\theta}=\pi/2, 0.1$ when $r/a=0.5$. For $\bar{\theta}=0$ when $r/a=0.5$, $J$ scales superdiffusively within our finite simulated lengths, and the scaling is $J \sim N^{-0.85}$.
• Figure 3: The profile of steady-state local temperature $T_n$ of $n^{\rm th}$ rotor of a nonlinear metamaterial of rotors of length $N=1024$. The six lines are for different equilibrium angle $\bar{\theta}$ and $r/a$ as given in the figure. Other parameters are $m=k=1, \gamma_L=\gamma_R=0.5, T_L=0.01,T_R=0.5$. The long-time average is taken over $7\times 10^7$ steps after the transient time of $7 \times 10^7$ steps.