Entanglement entropy of free fermions in timelike slices

We define the entanglement entropy of free fermion quantum states in an arbitrary space-time slice of a discrete set of points and particularly investigate timelike (causal) slices. For one-dimensional lattice free fermions with an energy bandwidth ${E}_{0}$, we calculate the time-direction entanglement entropy ${S}_{A}$ in a time-direction slice of a set of times ${t}_{n}=n\ensuremath{\tau}$ $(1\ensuremath{\le}n\ensuremath{\le}K)$ spanning a time length $t$ on the same site. For zero-temperature ground states, we find that ${S}_{A}$ shows volume law when $\ensuremath{\tau}\ensuremath{\gg}{\ensuremath{\tau}}_{0}=2\ensuremath{\pi}/{E}_{0}$; in contrast, ${S}_{A}\ensuremath{\sim}\frac{1}{3}lnt$ when $\ensuremath{\tau}={\ensuremath{\tau}}_{0}$, and ${S}_{A}\ensuremath{\sim}\frac{1}{6}lnt$ when $\ensuremath{\tau}<{\ensuremath{\tau}}_{0}$, resembling the Calabrese-Cardy formula for one flavor of nonchiral and chiral fermion, respectively. For finite-temperature thermal states, the mutual information also saturates when $\ensuremath{\tau}<{\ensuremath{\tau}}_{0}$. For noneigenstates, volume law in $t$ and signatures of the Lieb-Robinson bound velocity can be observed in ${S}_{A}$. For generic space-time slices with one point per site, the zero-temperature entanglement entropy shows a clear transition from area law to volume law when the slice varies from spacelike to timelike.