Observation of multiple attractors and diffusive transport in a periodically driven Klein-Gordon chain

We consider a Klein-Gordon chain that is periodically driven at one end and has dissipation at one or both boundaries. An interesting numerical observation in a recent study [Prem et al., Phys. Rev. B 107, 104304 (2023)] was that for driving frequency in the phonon band, there is a range of values of the driving amplitude $F_d\in (F_1,F_2)$ over which the energy current remains constant. In this range the system exhibits a traveling wave solution termed a "resonant nonlinear wave" (RNW). It was noted that the RNW mode occurs over a range $(F_1,F_2)$ and shrinks with increasing system size, $N$. Remarkably, we find that the RNW mode is in fact a stable solution even for $F_d>F_2$, and that in this regime there exist two attractors, both with finite basins of attraction. We improve the perturbative treatment for the RNW mode, presented in the earlier work, by including the contributions of third harmonics. We also consider the effect of thermal noise at the boundaries and find that the RNW mode is stable for small temperatures. Corresponding to the two attractors for large $F_d$ at zero temperature, the system can now be in two nonequilibrium steady states. Finally, we present results for a different driving protocol [Komorowski et al., Commun. Math. Phys. 400, 2181 (2023)] where $F_d$ is taken to scale with system size as $N^{-1/2}$ and dissipation is only at the nondriven end. We find that the steady state for this case can be characterized by Fourier's law. We point out interesting differences that occur because of our dynamics being nonlinear and Hamiltonian. Our results suggest the intriguing possibility of observing the high-current-carrying RNW phase in experiments by careful preparation of initial conditions.