Operator-Space Entanglement Entropy in Random Fermionic Gaussian Dynamics

Core claim

The paper's central claim is that for a Majorana string of length $M$ in $L$ Majorana modes evolved under Brownian SYK$_2$ dynamics, the thermodynamic-limit operator-space entanglement entropy density is exactly given for all times and all fractions $q=\ell/L$, $\theta=M/L$ by the one-dimensional quadrature $\lim_{L\to\infty} E[S_{op}(t)]/L = \int_0^\infty d\tau\,\{\theta(2-\frac{1}{1+\tau}) - \frac{1}{2}[\frac{\tau}{1+\tau}(h_- - a) + (h_+ - a)]\}$, where $h_\pm(\tau,t)$ are the two real branches of the Hamiltonian characteristic flow solving the resolvent PDE, and $a=1-2\theta$; general $q,\theta$ follow from the exact symmetry $S_{op}(q,\theta;t)=S_{op}(\theta,q;t)$. At the balanced point $q=\theta=1/2$ this reduces to $s_{op}(t)=2\log 2 - 1 - \sum_{k\ge 1} e^{-2kt} L^{(1)}_{2k-1}(4kt)/(2k^2(4k^2-1))$. In the stationary limit the same machinery yields the Gaussian Page curve $E[S_{op}(\ell)]/L = q\int_{\lambda_-}^{\lambda_+}\rho_*(\lambda)h_2(\lambda)\,d\lambda$, with $\rho_*$ the real Jacobi density and endpoints $\lambda_\pm=(\sqrt{\theta(1-q)}\pm\sqrt{q(1-\theta)})^2$; the balanced saturation value is $2\log 2 - 1$, set by the arcsine law. These curves form a family parameterized by $q$ and $\theta$ that differs from the Page curve of Haar-random states and from earlier pure-state Gaussian results. The paper further claims that operator and state entanglement share the same thermodynamic-limit evolution, while finite-size corrections differ and obey Eq. (52): a $1/L$ correction survives for the real ($\beta=1$) process describing the operator entropy but cancels for the complex ($\beta=2$) state process.