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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 S₀ in a time-direction slice of a set of times t₍=n (1) spanning a time length t on the same site. For zero-temperature ground states, we find that S₀ shows volume law when ₀=2/E₀; in contrast, S₀13lnt when =₀, and S₀16lnt when <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 <₀. For noneigenstates, volume law in t and signatures of the Lieb-Robinson bound velocity can be observed in S₀. 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.
Liu et al. (Thu,) studied this question.