Page Curves and Islands in Charged Dilaton Black Holes

Abstract: We study the Page curve in the four dimensional asymptotically flat eternal Garfinkle-Horowitz-Strominger dilaton black holes by applying the "quantum extremal surface" prescription. The results demonstrate that without island, the entanglement entropy of Hawking radiation is proportional to time and divergent at late times. While taking account of the emergence of the island extending slightly outside the event horizon, the entanglement entropy of Hawking radiation stops growing and eventually reaches a saturation value, which is twice of the Bekenstein-Hawking entropy and consistent with the finiteness of the von Neumann entropy of eternal black holes. Moreover, we discuss the impact of the parameter $n$ and charge $Q$ on the Page time. The influence of $n$ on it can neglected when the ratio of the charge $Q$ to the black hole mass $M$ is sufficiently small, but the charge $Q$ has a significant impact on Page time. Importantly, at the extremal case, the Page time becomes divergent or vanishing, in which the semiclassical theory needs further investigation.