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 E0, we calculate the time-direction entanglement entropy SA in a time-direction slice of a set of times tn=nτ (1≤n≤K) spanning a time length t on the same site. For zero-temperature ground states, we find that SA shows volume law when τ≫τ0=2π/E0; in contrast, SA∼13lnt when τ=τ0, and SA∼16lnt when τ<τ0, 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 τ<τ0. For noneigenstates, volume law in t and signatures of the Lieb-Robinson bound velocity can be observed in SA. 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.