Two-parameter Family-Vicsek scaling in a dissipative XXZ spin chain

Abstract

Family-Vicsek (FV) scaling provides an understanding for the growth and finite-size saturation of fluctuations in classical systems. Here, we extend the FV roughness to transferred segment magnetization after quantum quenches in a dissipative XXZ spin chain with homogeneous gain and loss, starting from a nonequilibrium steady state with finite magnetization. In the non-interacting limit, we derive a closed-form expression for the roughness in the presence of dissipation. It displays two-parameter FV scaling and smoothly interpolates between the clean ballistic behavior and the dissipation dominated scalings. For interacting chains, tensor-network simulations show that the non-dissipative ballistic growth at finite magnetization is robust, whereas the full Lindblad evolution is generically controlled by the dissipative relaxation time and exhibits a dissipation-dominated collapse.

Figures (10)

• Figure 1: (a) FV scaling collapse for the unitary evolution in the XX limit ($\Delta=0$) starting from the NESS with $\zeta=0.5$ ($1/3$-filling, $\bar{n}=1/3$). The symbols show the numerical TEBD data for $W^{(H)}(\ell,t)$ while the solid lines show the analytic result from Eq. \ref{['eq:kappa2_xx_finiteGamma']} for the same parameters. Dashed lines indicate the universal functions given in Eq. \ref{['eq:fH_asymptotics']}. The crossover time $t^*$ is estimated from Eq. \ref{['eq:tstar_unitary']} and indicated by the vertical arrow. (b,c) FV scaling collapse for the full Lindblad evolution for weak (b), ($\gamma_l=0.02J$) and strong (c) ($\gamma_l=0.15J$), dissipation respectively, and $\zeta=0.5$ as before. The roughness saturates already at $t\sim t_{\Gamma}$. The crossover time $t_{\Gamma}$ is indicated by the vertical arrow. System sizes are $L=500$ and segment lengths $\ell=10, 20,40,80, 160$. In the TEBD simulations, we used a maximum bond dimension $\chi_{\rm max}=64$ and a time step $dt=0.01/J$. The inset shows the same unrescaled data, on a log-log scale, where the three regimes of microscopic transient, FV growth, and FV saturation are clearly visible.
• Figure 2: (a) FV scaling collapse for the unitary evolution in the interacting XXZ chain with $\Delta=\{0.2, 1.0, 2.0\}$ and $\zeta=0.1$. The symbols show the numerical TEBD data while the dashed lines are the asymptotic forms of the universal function corresponding to the growth $\sim t^{\beta}$. (b) same as in (a) but for the Lindbladian evolution with $\gamma_l=0.15 J$ and $\zeta=0.1$. System sizes are $L=500$ and segment lengths $\ell=10, 20,40,80, 160$. In the TEBD simulations, we use maximum bond dimension $\chi_{\rm max}=32$ and a time step $dt=0.01/J$.
• Figure 3: FV scaling collapse for the unitary evolution in the XX limit ($\Delta=0$) starting from the NESS with $\zeta=0.5$ ($1/3$-filling, $\bar{n}=1/3$). The symbols show the numerical TEBD data for $W^{(H)}(\ell,t)$ while the solid lines show the analytic result from Eq. \ref{['sm:eq:mu2_unitary_bessel']} for the same parameters. The crossover time $t^*$ is estimated from Eq. \ref{['sm:eq:tstar_unitary']} and indicated by the vertical arrow. The universal function is shown as dashed lines, with the growth and saturation regimes corresponding to Eqs. \ref{['sm:eq:W_unitary_growth']} and \ref{['sm:eq:W_unitary_sat']}. System sizes are $L=1000$ and segment lengths $\ell=10, 20,40,80, 160$ and in the TEBD simulations we used a maximum bond dimension $\chi_{\rm max}=64$ and a time step $dt=0.01/J$. The agreement between the numerical data and the analytic result is excellent, confirming the validity of the FV scaling collapse and the extracted exponents. The inset shows the same unrescaled data, on a log-log scale, where the three regimes of microscopic transient, FV growth, and FV saturation are clearly visible.
• Figure 4: Scaling collapse in the dissipation-dominated regime $t_{\Gamma}\sim \Gamma^{-1}\ll t^*$, where the roughness saturates already at $t\sim t_{\Gamma}$ while the ballistic scaling variable $x=Jt/\ell$ remains small. The crossover time $t_{\Gamma}$ is indicated by the vertical arrow. The solid lines are the analytic result from Eq. \ref{['sm:eq:mu2_lindblad_bessel']} while the symbols show the numerical TEBD data for $W^{(\mathcal{L})}(\ell,t)$ for the same set of parameters. System size is fixed to $L=500$ and $\zeta = 0.5$. The left panel corresponds to $\Gamma=0.02J$, while the right panel corresponds to $\Gamma=0.15J$. The vertical marker at $t_{\Gamma}$ highlights the crossover from the initial growth regime to saturation, which occurs at a time scale set by the dissipation rate rather than by the segment length. The solid lines show the analytic result from Eq. \ref{['sm:eq:mu2_lindblad_bessel']}, which captures the full time dependence of the roughness and agrees well with the numerical data. The insets represent the unrescaled data, indicating that the three regimes clearly visible in the unitary evolution are no longer present, as the short time transient is now followed by a direct crossover to saturation at $t\sim t_{\Gamma}$. System sizes are $L=500$ and segment lengths $\ell=10, 20,40,80, 160$ and in the TEBD simulations we used a maximum bond dimension $\chi_{\rm max}=32$ and a time step $dt=0.01/J$.
• Figure 5: FV scaling collapse for the unitary evolution in the interacting XXZ chain with $\Delta=\{0.2, 1.0, 2.0\}$ and $\zeta=0.1$. The symbols show the numerical TEBD data while the dashed lines are the asymptotic forms of the universal function corresponding to the growth $\sim t^{\beta}$. System sizes are $L=500$ and segment lengths $\ell=10, 20,40,80, 160$ and in the TEBD simulations we use maximum bond dimension $\chi_{\rm max}=32$ and a time step $dt=0.01/J$.