Eigenstate capacity and Page curve in fermionic Gaussian states

Capacity of entanglement (CoE), an information-theoretic measure entanglement, defined as the variance modular Hamiltonian, is known to capture deviation from maximal entanglement. We derive exact expression for average eigenstate CoE in fermionic Gaussian states a finite series, valid arbitrary bi-partition total system. Further, we consider complex SYK$_2$ model thermodynamic limit and obtain closed-form CoE. In this limit, becomes independent system size. Moreover, when subsystem size half system, leading volume-law coefficient approaches value $\pi^{2}/8 - 1$. identify distinguishing feature between integrable quantum-chaotic systems. confirm our analytical results by numerical computations.