Anomalies of the Entanglement Entropy in Chiral Theories

TL;DR

The paper demonstrates that entanglement entropy in theories with gravitational or mixed gauge-gravitational anomalies is frame-dependent, exhibiting a universal anomaly under diffeomorphisms or gauge transformations. It develops multiple derivations in 2d, 4d, and 6d, relating the entropy variation to index theorems, anomaly coefficients, and boundary-bulk (Hall phase) constructions, with explicit local expressions for the entropy shift concentrated on the entangling surface. A central insight is that the anomaly is captured by low-energy zero-mode spectra and index theorems, and can be probed via twist flux backgrounds and boundary shear in Hall phases. The results have broad implications for how entanglement is defined in chiral theories and suggest observable signatures in condensed matter analogs and potentially the Standard Model, motivating further exploration of higher-dimensional anomalies and their entanglement fingerprints.

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.

Figures (5)

• Figure 1: Two different Cauchy slices $\Sigma$ and $\Sigma'$ of the same domain of dependence $D[A]$, connected by a diffeomorphism $\xi$.
• Figure 2: Example of mutual information regulator. All intervals are understood to be at the same time slice.
• Figure 3: Zoomed in view near one of the endpoints of the interval, demonstrating transformation of the cutoff under a local boost. $(u_1, v_1)$ and $(u_1', v_1')$ refer to the lengths of the cutoff along the $(u,v)$ directions before and after the boost respectively.
• Figure 4: Boundary region $A$ and associated bulk region $m_A$. Under the described shear operation of the full metric, the interval on the boundary is distorted as shown.
• Figure 5: Weyl fermion studied on $\mathbb{R}^{1,1} \times T^2$ with magnetic flux on $T^2$. Lowest energy modes have spin aligned with magnetic field, and chiral nature of Weyl fermion means that velocity is anti-aligned with spin: thus the low energy physics is that of a chiral CFT$_2$. We study the entanglement entropy of a region $A$ that is a product of an interval on $\mathbb{R}^{1,1}$ and $T^2$.