Inhomogeneous quenches and GHD in the $ν= 1$ QSSEP model

Abstract

We investigate the dynamics of the $ν=1$ Quantum Symmetric Simple Exclusion Process starting from spatially inhomogeneous initial states. This one-dimensional system of free fermions has time-dependent stochastic hopping amplitudes that are uniform in space. We focus on two paradigmatic setups: domain-wall melting and the expansion of a trapped gas. Both are investigated by extending the framework of quantum generalized hydrodynamics to account for the underlying stochastic dynamics. We derive the evolution of the local quasiparticle occupation function, which characterizes the system at large space-time scales, and analyze the resulting entanglement spreading. By incorporating quantum fluctuations of the occupation function and employing conformal field theory techniques, we obtain the exact contribution to the entanglement entropy for each individual noise realization. Averaging over these realizations then yields the full entanglement statistics in the hydrodynamic regime. Our theoretical predictions are confirmed by exact numerical calculations. The results presented here constitute the first application of quantum generalized hydrodynamics to stochastic quantum systems, demonstrating that this framework can be successfully extended beyond purely unitary dynamics to include stochastic effects.

Figures (7)

• Figure 1: Sketch of the time evolution of the local occupation function $n_k(x, t)$ after a domain-wall quench in the $\nu=1$ QSSEP. The dynamics of $n_{k}(x,t)$ follows the stochastic differential equation \ref{['eq:SDE_wigner']}. Colored regions correspond to $n_k(x, t)=1$ for single noise realizations at different times. We compute the statistical properties of the out-of-equilibrium entanglement entropy $S_{\ell}(t)$ between two intervals $A$ and $B$, connected at $x = \ell$. The exact contribution to entanglement of each noise realization is obtained using QGHD. In this framework, the quantum fluctuations of $n_k(x, t)$ are introduced in the form of a free massless boson living on the Fermi contour $\Gamma_{t}$ that separates the fully filled and fully empty regions in the $(x,k)$ plane. The contribution to the entanglement entropy of $A$ of each noise realization is given at, e.g., time $t = t_2$ by the correlation function of twist fields inserted at the Fermi points: the intersection of the line $x=\ell$ with the Fermi contour $\Gamma_{t_2}$ (black dots).
• Figure 2: Left panel: Profile of the average entanglement entropy for the subsystem $A = [-L/2,\ell]$ at different times $t$ starting from the domain-wall state \ref{['eq:DWdef']} in the $\nu=1$ QSSEP. The symbols are the exact average entanglement entropy over $\sim 10^3$ noise realizations, computed numerically in a lattice of size $L=160$. The errorbar is estimated from the standard deviation of the mean. The dashed curves correspond to the analytical prediction from QGHD in Eq. \ref{['eq:avgSx0int']}. Right panel: Same as in the left panel, but the symbols correspond to the exact numerical results after subtraction of the first term in Eq. \ref{['eq:avgSx0int']}, in order to extract the scaling function $\mathcal{S}(\ell/\sqrt{t})$. The data at different times all collapse onto our prediction for $\mathcal{S}(\ell/\sqrt{t})$ in Eq. \ref{['eq:scal_fun_S']} (black dashed line), clearly indicating diffusive spreading of the entanglement.
• Figure 3: Time evolution of the average half-system entanglement in the $\nu=1$ QSSEP \ref{['eq:model_dH']} starting from the domain wall state \ref{['eq:DWdef']}. The symbols correspond to the exact average value computed over $\sim 10^3$ noise realizations for increasing total system size $L$. The errorbar is estimated from the standard deviation of the mean. The continuous black curve corresponds to the analytical prediction in Eq. \ref{['eq:halfsysS']} derived using QGHD methods. The top inset shows the same data with logarithmic $x$-axis for a better visualization of the logarithmic growth of the average entanglement entropy with time, expected from Eq. \ref{['eq:halfsysS']}. The bottom inset shows the time dependence of the standard deviation $\sigma_S$ of the half-system entanglement entropy. The exact numerical data (symbols) tend to the QGHD prediction in Eq. \ref{['eq:rel_fluct']}, confirming that the relative fluctuations of entanglement entropy vanish as $t\to \infty$.
• Figure 4: Sketch of the time evolution of the local occupation function $n_k(x,t)$ in the free-expansion protocol studied in Sec. \ref{['sec:freexpsec']}. The colored regions correspond to $n_k(x,t) = 1$. In the left panel, we show the occupation function of the initial state, corresponding to the ground state of Eq. \ref{['eq:Ham_beta']} at finite $\beta$. The Fermi contour $\Gamma_t$ evolves stochastically in time according to Eq. \ref{['eq:SDE_wigner']}. In the right panel, we show it for several noise realizations at time $t$. The average entanglement entropy $S_{\ell}(t)$ for $A = (-L/2,\ell]$ is obtained by averaging over the contribution of all possible Fermi contours. Each contribution is determined by the Fermi points: the intersections of the system cut $x=\ell$ with $\Gamma_t$. In the right panel, we highlight a noise realization with $4$ Fermi points, a scenario that does not occur in the domain-wall melting.
• Figure 5: Average half-system entanglement entropy $\langle S_{\ell = 0}(t) \rangle$ as a function of time in the free expansion protocol, starting from the ground state of the Hamiltonian \ref{['eq:Ham_beta']} with $\beta = 0.25$. The symbols are exact average entropy over $\sim 10^3$ noise realizations. The corresponding errorbar is the standard deviation of the mean. We show results for increasing system sizes $L$, up to times in which the effects of boundaries at $x=\pm L/2$ are negligible. The solid curve is the QGHD prediction. Top inset: Same data as in the left panel but in logarithmic time scale. After times of order $\sim 10$, the exact numerical results tend to the QGHD prediction. Bottom inset: time dependence of the standard deviation $\sigma_S$ of the half-system entanglement entropy. Symbols are the exact numerical data and the solid curve is the QGHD prediction.