Signatures of global symmetry violation in relative entropies and replica wormholes

TL;DR

The paper addresses the absence of exact global symmetries in quantum gravity with black holes and proposes a quantitative diagnostic using relative entropy between Hawking radiation states and their symmetry-transformed counterparts. By combining the quantum extremal surface/island prescription with replica wormholes, the authors show that these relative entropies can be computed semiclassically and yield $\mathcal{O}(1)$ contributions after the Page time. They provide an explicit calculation for a $U(1)$ global symmetry of Dirac fermions and show that replica wormholes facilitate charge flow through the wormhole, connecting to the Page curve and Hayden–Preskill-type scrambling. The results also delineate the distinction between global and gauge symmetries in gravity and suggest broader implications for ensemble interpretations and potential experimental probes via Rényi relative entropies.

Abstract

It is widely believed that exact global symmetries do not exist in theories that admit quantum black holes. Here we propose a way to quantify the degree of global symmetry violation in the Hawking radiation of a black hole by using certain relative entropies. While the violations of global symmetry that we consider are non-perturbative effects, they nevertheless give $\mathcal{O}(1)$ contributions to the relative entropy after the Page time. Furthermore, using "island" formulas, these relative entropies can be computed within semi-classical gravity, which we demonstrate with explicit examples. These formulas give a rather precise operational sense to the statement that a global charge thrown into an old black hole will be lost after a scrambling time. The relative entropies considered here may also be computed using a replica trick. At integer replica index, the global symmetry violating effects manifest themselves as charge flowing through the replica wormhole.

Figures (9)

• Figure 1: (a) A zero temperature black hole in JT gravity coupled to bath. It has a nonperturbative description in terms of a boundary quantum mechanical system coupled to a CFT in half infinite space. (b) When we compute the entropy of a region $R$ in the bath that is large enough, there will be an island $I$ in the gravity region.
• Figure 2: The relative entropy in the free fermion model, as a function of the size of the region $R$ at fixed non-zero $\alpha$. The horizontal axis is $\ell(R)$ the length of $R$ in units of $\phi_r/c$ ($\ell(R) \equiv c(b_2 - a_2)/\phi_r \approx c b_2/\phi_r$). In the semi-classical approximation, there is a discontinuity in the relative entropy at the "Page length" $\ell_P (R)\sim e^{12S_0/c}$ when an island appears in the gravity region, see (\ref{['relresult1']}). It might seem that the relative entropy is large because it has $S_0$ in the exponential, but recall that in this model the ratio $S_0/c$ is kept finite and not very large Almheiri:2019yqk, so the relative entropy is $\mathcal{O}(1)$.
• Figure 3: (a) We consider two virtual processes $\ell_1$ and $\ell_2$ where charged particles run in loops. Only the larger loop $\ell_1$ contributes to the bulk relative entropy (as well as the entanglement between $I$ and $R$). (b) We imagine the decay of a neutral particle into 2 fermions at $x_1$ and $x_2$. Only the decay at $x_1$ contributes significantly to the bulk relative entropy. Note also that in an evaporating black hole, a morally similar picture would also predict that the relative entropy increases after the Page time because each Hawking mode will be entangled with a mode of opposite charge behind the horizon.
• Figure 4: A two-sided black hole in JT gravity coupled to two bath regions Almheiri:2019qdq. When we compute the entropy of a large region $R$ in the bath, depending on the parameters, we can have an island $I$ in the gravity region. At sufficiently late times $t>0$ there will always be an island.
• Figure 5: (a) The replica wormhole geometry $\mathcal{M}_2$ for computing $Z_2 (0)$. (b) For $Z_2 (\alpha)$, we have the symmetry transformation operators inserted around the cuts in one of the replicas.