Abstract
We study entanglement entropy in theories with gravitational or mixed U(1) gauge-gravitational anomalies in two, four and six dimensions. In such theories there is an anomaly in the entanglement entropy: it depends on the choice of reference frame in which the theory is regulated. We discuss subtleties regarding regulators and entanglement entropies in anomalous theories. We then study the entanglement entropy of free chiral fermions and self-dual bosons and show that in sufficiently symmetric situations this entanglement anomaly comes from an imbalance in the flux of modes flowing through the boundary, controlled by familiar index theorems. In two and four dimensions we use anomalous Ward identities to find general expressions for the transformation of the entanglement entropy under a diffeomorphism. (In the case of a mixed anomaly there is an alternative presentation of the theory in which the entanglement entropy is not invariant under a U(1) gauge transformation. The free-field manifestation of this phenomenon involves a novel kind of fermion zero mode on a gravitational background with a twist in the normal bundle to the entangling surface.) We also study d-dimensional anomalous systems as the boundaries of d + 1 dimensional gapped Hall phases. Here the full system is non-anomalous, but the boundary anomaly manifests itself in a change in the entanglement entropy when the boundary metric is sheared relative to the bulk.
Highlights
In two and four dimensions we use anomalous Ward identities to find general expressions for the transformation of the entanglement entropy under a diffeomorphism. (In the case of a mixed anomaly there is an alternative presentation of the theory in which the entanglement entropy is not invariant under a U(1) gauge transformation
We study the entanglement entropy of free chiral fermions and self-dual bosons and show that in sufficiently symmetric situations this entanglement anomaly comes from an imbalance in the flux of modes flowing through the boundary, controlled by familiar index theorems
We studied the structure of entanglement entropy in quantum field theories with gravitational anomalies or mixed gauge-gravitational anomalies
Summary
A more familar example of an anomaly in the entanglement entropy comes in CFT’s. Here, scale invariance maps a spatial region A to a rescaled region A , for example one twice as large. This is not so disturbing if we think of the CFT as an effective field theory description of a microscopic theory which has a shortest distance scale This non-scale invariance of the entanglement entropy is closely related to the trace anomaly, which is a nonzero trace T of the stress tensor which arises when a CFT is quantized on a curved spacetime. This anomaly exists in even numbers of dimensions, e.g. in 4 dimensions the trace anomaly (of a theory without a diffeomorphism anomaly) takes the form. The following analogy obtains: trace anomaly: log divergence of S :: chiral anomaly: boost non-invariance of S. (1.6)
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