Nonequilibrium Steady State Generated by a Moving Defect: The Supersonic Threshold

We consider the dynamics of a system of free fermions on a 1D lattice in the presence of a defect moving at constant velocity. The defect has the form of a localized time-dependent variation of the chemical potential and induces at long times a nonequilibrium steady state (NESS), which spreads around the defect. We present a general formulation that allows recasting the time-dependent protocol in a scattering problem on a static potential. We obtain a complete characterization of the NESS. In particular, we show a strong dependence on the defect velocity and the existence of a sharp threshold when such velocity exceeds the speed of sound. Beyond this value, the NESS is not produced and, remarkably, the defect travels without significantly perturbing the system. We present an exact solution for a $\delta$-like defect traveling with an arbitrary velocity and we develop a semiclassical approximation that provides accurate results for smooth defects.

Figure 1

Fermion density generated by a $\delta$-like defect [$V(x)=c\delta (x)$, $c=0.5$] moving at $v=0.3$ (top) and $v=1.5$ (bottom). The defect is initially placed at zero and it moves along the line dashed in red. In the subsonic case the constant values along the rays indicate the realization of a LQSS, which is instead absent in the supersonic regime.

Figure 2

The numeric fermionic density $⟨d_j^{†}{d}_j⟩$ with $j=(v+\zeta )t$ as a function of $\zeta$ at three different times is tested against the analytic LQSS for a $\delta$-like defect $V(x)=c\delta (x)$ with $c=0.5$ and velocity $v=0.3$. The density is fixed by $k_f=\pi /3$. The oscillations mentioned in the main text are smeared out by averaging on neighboring sites.

Figure 3

Analytic prediction for the excitation density profile propagating on the left (left panel) and on the right (right panel) of a $\delta$-like defect. The dashed red line is the initial Fermi sea, the black line the spreading excitation whose integral (shaded area) equals the spatial fermionic density, which jumps around the defect (see Fig. 2). The same parameters of Fig. 2 are used.

Figure 4

The numerical fermionic density is tested against the semiclassical LQSS for a Gaussian shaped repulsive potential $V(x)=e^{-\sigma x^2}$ with $\sigma =0.04$, $k_f=\pi /2$. Top: subsonic defect ($v=0.3$) and $t=636$. The defect is placed where the discontinuity occurs and a LQSS is produced. Bottom: supersonic defect ($v=1.5$, $t=388$) and placed in correspondence with the rightmost peak: the LQSS is indeed absent and the semiclassical prediction captures the density profile on the defect (inset). The corrections to the analytic prediction are a combined effect of (i) the particles initially sat on the defect that have not yet managed to spread and (ii) the delay time experienced by the particles scattering on the defect. Both these effects are negligible at late time and in the scaling limit. Quantum effects can be recognized in the propagating ripples [119, 120, 121], more evident in the supersonic case.