Dynamics of number entropy for free fermionic systems in presence of defects and stochastic processes

TL;DR

This work analyzes the dynamics of number entropy (NE) in a 1D free-fermion chain with defects and stochastic processes. By leveraging particle-number conservation, NE is computed from the distribution $p(n,t)$ of particles in a subsystem, which is determined by the eigenvalues $f_m(t)$ of the subsystem correlation matrix; for a conformal defect, $p(n,t)$ becomes a shifted binomial and NE grows as $\sim \frac{1}{2}\log t$ in the thermodynamic limit, revealing a direct link between NE and defect scattering through reflection $R(k)$ and transmission $T(k)$. When stochastic processes—Stochastic Unitary Processes (SUP) and Quantum State Diffusion (QSD)—are added, NE scales as $\ln t$ (SUP) and shows hints of $\ln\ln t$ (QSD), illustrating that NE grows logarithmically slower than von Neumann entanglement entropy across these settings. Overall, the study uncovers a robust relationship between eigenvalue dynamics, scattering data, and slow NE growth, suggesting a universal mechanism for charge-based entanglement spreading in non-interacting and noisy quantum systems.

Abstract

We investigate the dynamics of number entropy in a chain of free fermions subjected to both defects and stochastic processes. For a special class of defects, namely conformal defects, we present analytical and numerical results for the temporal growth of number entropy, the time evolution of the number distribution, and the eigenvalue profile of the associated correlation matrix within a subsystem. We show that the number entropy exhibits logarithmic growth in time, originating from the Gaussian structure of the number distribution. We find that the eigenvalue dynamics reveal a profound connection to the reflection and transmission coefficients of the associated scattering problem for a broad range of defects. When stochastic processes are introduced, specifically Stochastic Unitary Processes (SUP) and Quantum State Diffusion (QSD), the number entropy scales as $\ln(t)$ in the SUP case and shows strong hints of $\ln [\ln(t)]$ scaling in the QSD case. These findings establish compelling evidence that number entropy grows logarithmically slower than the corresponding von Neumann entanglement entropy across a wide class of systems.

Figures (8)

• Figure 1: NE [$S_N(t)]$ as a function of time [$t$] for system size $L_s = 40,80$, and $160$ with $g = 0.5$ and $g_c = 0.3$. The solid line shows the exact numerical results. The blue circled line shows the analytically obtained NE given by Eq. \ref{['eq:S_Nconformal']}. The black dashed line is given [Eq. \ref{['eq:sinf']}] by $S_{N\infty}(t) = \frac{1}{2}\log_2 [2\pi e\,v\,t \,\lambda^2(1-\lambda^2)]$, where $v = 2g/\pi$ and $\lambda = g_c/g$. The analytical result and $S_{N\infty(t)}$ are consistent with the numerical solution up to time $L_s/g$. The inset shows $\frac{dS_N}{dt}$ as a function of time for numerically obtained $S_N(t)$ with $L_s = 160$, which is represented by the solid orange line in the main plot. The vertical dashed lines mark the time when non-analyticity is observed. Red vertical dashed line corresponds to $t = 320$ and blue vertical dashed line corresponds to $t = 640$.
• Figure 2: (a) Eigenvalue profile for the system part of the correlation matrix (obtained by ordering the eigenvalues in descending order) at different times for $g_c = 0.3$, $g = 0.5$, $L_s = 160$, $L = 2000$. As the system is quenched, the values of the eigenvalue decrease from 1 and saturate at $(1-\lambda^2)$, where $\lambda = g_c/g$. It can be seen that at times $t = 320$ (blue) and $t = 640$ (purple), there is a transition in the values. These times correspond to $t = nL_s/g$ for $n \in \mathbb{N}$. The saturation values are given by $(1-\lambda^2)^n$ for $n = 1, 2$ and $3$. (b) Eigenvalue profile for the system part of the correlation matrix for $g_c = 0.2,0.3,0.4$ with $L_s = 160$, $g=0.5$, and at a given time snapshot $t = 200$. The black dashed lines mark the saturation value of the eigenvalue profile given by $G(\lambda) = 1-\lambda^2$, where $\lambda = g_c/g$. The vertical red line marks the position of the front, which is given by $(L_s - v\,t)$ where $v = 2g/\pi$ and is independent of $g_c$.
• Figure 3: Evolution of the PDF of the particle number $[p_C(n,t)]$ in the system for the setup with a conformal defect, starting with domain wall initial condition. The solid lines are the numerically obtained distribution from the correlation matrix. The black dashed lines are obtained by plotting Eq. \ref{['eq:p_conform']} for the corresponding time values. Here, $L_s = 160$, $g = 0.5$ and $g_c = 0.4$.
• Figure 4: $S_N(t)$ vs $t$ when the probes are present on the system sites for different system sizes. The data is averaged over 100 Quantum Trajectories. The solid lines show the numerically obtained data, the blue dashed line shows the early time scaling, and the black dashed line shows late time scaling. The functions plotted are $\mathcal{F}_n(t) = a_n\ln(t) + b_n$ for $n=1,2,3$ and $\mathcal{G}(t) = a_4\,\ln [\ln(t)] + b_4 \ln(t) + c$. (a) Probes under SUP with $\gamma = 0.1$, $g_c = 0.4$ and $g = 0.5$. Two logarithmic scalings with time, given by $\mathcal{F}_1(t)$ and $\mathcal{F}_2(t)$, are observed. $a_1 = 0.62$, $b_1 = 0.12$, $a_2 = 0.38$ and $b_2 = 1.26$. (b) Probe under QSD protocol with $\gamma = 0.05$, $g_c = 0.1$ and $g = 0.5$. At early times, logarithmic growth with time, marked by $\mathcal{F}_3(t)$, is observed. At later times, the growth slows down to a form involving a double logarithmic scaling with time, which is given by $\mathcal{G}(t)$. $a_3 = 0.66$, $b_3 = -0.61$, $a_4 = 5.32$, $b_4 = -0.71$ and $c = -2.46$.
• Figure S1: Evolution of the eigenvalue profile of the system part of the correlation matrix for a homogeneous setup with no defect [Eq. \ref{['supp_eq:Ham']} with $g = g_c$]. Here, we have chosen $L_s = 160$ and $g = 0.5$. As clearly seen in the figure, among the $L_s$ eigenvalues, only $\mathcal{O}(1)$ of them are different from 1 and 0, which contribute to entanglement.