Dissipation induced elastic-mode instability with topological excitation in holographic non-equilibrium steady cnoidal wave supersolid

TL;DR

The paper tackles the stability of cnoidal-wave supersolids in non-equilibrium, finite-temperature settings by combining a dissipative Gross-Pitaevskii framework with a finite-temperature holographic superfluid model. It identifies the elastic Goldstone mode, arising from translational symmetry breaking, as the key driver of dynamical instability, linking its growth to the nucleation of topological soliton–antisoliton flips that change the winding number and relax the system toward a homogeneous superfluid. In strong dissipation, the elastic-mode instability dominates over thermodynamic instability, with nonlinear evolution revealing explicit phase-slip events and topological transitions consistent with Landau-type instabilities. Overall, the work demonstrates that cnoidal-wave supersolids are dynamically unstable at finite temperature and highlights the utility of holography for capturing dissipative, topologically nontrivial non-equilibrium states and their relaxation pathways.

Abstract

The possible existence of an exotic phase of matter rigid like a solid but able to sustain persistent and dissipation-less flow like a superfluid, a "supersolid", has been the subject of intense theoretical and experimental efforts since the discovery of superfluidity in Helium-4. Recently, it has been proposed that nonlinear periodic modulations known as cnoidal waves, that naturally emerge in Bose-Einstein condensates, provide a promising platform to find and study supersolidity in non-equilibrium phases of matter. Nevertheless, so far the analysis has been limited to a one-dimensional zero-temperature system. By combining the dissipative Gross-Pitaevskii equation with a finite temperature holographic model, we show that the proposed cnoidal wave supersolid phases of matter are dynamically unstable at finite temperature. We ascribe this instability to the dynamics of the "elastic" Goldstone mode, which arises as a direct consequence of translational order in the presence of dissipation, and establish a direct connection between the elastic-mode instability of the supersolid state and the nucleation of topological excitations during the relaxation towards a homogeneous equilibrium state, which resembles the Landau instability in superfluids. Finally, we numerically confirm the dominant role of the elastic-mode instability in the collision between cnoidal waves in the strong dissipation limit.

Figures (16)

• Figure 1: Cnoidal waves.(a) Near-cnoidal wave trains in the Atlantic ocean photographed from the Whale lighthouse (France), Images are taken from https://en.wikipedia.org/wiki/Cnoidal_wave. (b) A cnoidal wave, characterised by sharper crests and flatter troughs than in a sine wave.
• Figure 2: Inhomogeneous superflow states (cnoidal waves) of the GP equation. Profile of condensate density $\rho(x)$ with Bloch-wave vector $k=0.5k_0$ for dark-soliton, $k=1.11k_0$ for linear wave and $k=1.25k_0$ for uniform solutions. The green line is a fitting to linear wave solution Eq. \ref{['linfit']} with form $\rho=0.616-0.0573\cos(0.785x)$ while the blue line is the dark-soliton fitting function of Eq. \ref{['darkfit']} with form $\rho=0.997\tanh(1.0021x)^2$.
• Figure 3: Linear stability of cnoidal wave supersolidity from GP equation. Dispersion relation of sound modes $\omega_s^{\pm}$ and elastic modes $\omega_e^{\pm}$ on top of the cnoidal wave solutions of the 1D GP equation, Eq.\ref{['GP']}. Panels (a) and (b) are for $k=1/2k_0,2/3k_0$ and $\eta=0$. Insets display zoomed-in views of modes $\omega_e^\pm$. Panels (c) and (e) are the real and imaginary parts of the dispersion relations for $k=1/2k_0$ with $\eta=0.01$ and panels (d) and (f) are the real and imaginary parts of the dispersion relations for $k=2/3k_0$ with $\eta=0.01$.
• Figure 4: Cnoidal wave solutions in a finite temperature holographic model.(a) Amplitude of complex bulk scalar field $\psi(z,x)$ with $k=1.5k_0$. (b) Profile of the condensate density for a dark-soliton solution ($k=0.5k_0$), linear-wave solution ($k=1.8k_0$) and a uniform solution ($k=2k_0$), respectively. The blue line is fitting function of dark-soliton solution with $\frac{|\Psi_1|^2}{T_c^2}=0.206\tanh(1.633x)^2$ and green lines is fitting function of linear-wave function with $\frac{|\Psi_1|^2}{T_c^2}=0.111-0.056\cos(1.5708x-0.821)$.
• Figure 5: Supersolid excited states in the holographic model. Total current density $j_x$(a) and chemical potential $\mu$(b) as a function of the superflow parameter $k$. (c) Free energy as a function of $\mu$ for inhomogeneous and homogeneous solutions with superflow. The orange symbols indicate the location of the solutions explicitly shown in Fig. \ref{['states']}, where circle is the dark-soliton solution, triangle is the linear-wave solution and square is a uniform solution.