  We study entanglement entropy for parity-violating (time-reversal breaking)
quantum field theories on $\mathbb{R}^{1,2}$ in the presence of a domain wall
between two distinct parity-odd phases. The domain wall hosts a 1+1-dimensional
conformal field theory (CFT) with non-trivial chiral central charge. Such a CFT
possesses gravitational anomalies. It has been shown recently that, as a
consequence, its intrinsic entanglement entropy is sensitive to Lorentz boosts
around the entangling surface. Here, we show using various methods that the
entanglement entropy of the three-dimensional bulk theory is also sensitive to
such boosts owing to parity-violating effects, and that the bulk response to a
Lorentz boost precisely cancels the contribution coming from the domain wall
CFT. We argue that this can naturally be interpreted as entanglement inflow
(i.e., inflow of entanglement entropy analogous to the familiar Callan-Harvey
effect) between the bulk and the domain-wall, mediated by the low-lying states
in the entanglement spectrum. These results can be generally applied to 2+1-d
topological phases of matter that have edge theories with gravitational
anomalies, and provide a precise connection between the gravitational anomaly
of the physical edge theory and the low-lying spectrum of the entanglement
Hamiltonian.
