Dynamics of entanglement entropy and particle number distribution in disordered, free fermionic systems

The information contained in a quantum state $\rho$ is quantified by the entanglement entropy $S=-\textrm{tr}(\rho \ln \rho)$, which is difficult to measure. For systems with particle number conservation, $S$ is the sum of the number entropy, $S_N$, and the configuration entropy, $S_{conf}$, which have been measured recently in a cold-gas experiment [1]. We here show that for systems of non-interacting fermions, including the case of disorder, the time evolution of the second Renyi entropy $S^{(2)}=-\ln\textrm{tr}(\rho^2)$ is determined by the exponent of corresponding number entropies. As a consequence in free fermionic systems a dynamical growth of entanglement is always related to a slower growth of the number entropy. We numerically illustrate this for different tight-binding fermionic models including the case of off-diagonal disorder for which the entanglement entropy shows an ultra slow, double logarithmic growth in time and give an outlook to interacting systems showing many-body localization.\\ [1] A. Lukin, \textit{et al.} \textit{Probing entanglement in a many-body localized system}, Science \textbf{364}, 256 (2019).