Large-Charge Limit of AdS Boson Stars with Mixed Boundary Conditions

It was recently shown that charged AdS boson stars can reproduce the universal structure of the lowest scaling dimension in the subsector of a CFT with fixed large global $U(1)$ charge $Q$. Using the model consisting of Einstein-Maxwell gravity with a negative cosmological constant, coupled to a $U(1)$-charged conformally massless scalar with the fourth-order self interaction, we construct a class of charged AdS boson star solutions in the large $Q$ limit, where the scalar field obeys a mixed boundary condition, parameterized by $k$ that interpolates between the Neumann and Dirichlet boundary conditions corresponding to $k=0$ and $\mathrm{\infty }$ respectively. By varying $k$, we numerically read off the $k$ dependence of the leading coefficient $c_{3\mathrm{/}2}(k)\equiv {\lim }_{Q\to \mathrm{\infty }}M\mathrm{/}{Q}^{3\mathrm{/}2}$. We find that $c_{3\mathrm{/}2}(k)$ is a monotonously increasing function which grows linearly when $k$ is sufficiently small. When $k\to \mathrm{\infty }$, $c_{3\mathrm{/}2}(k)$ approaches the maximal value at a decreasing rate given by $k^{-3\mathrm{/}2}$ . We also obtain a close form expression that fits the numerical data for the entire range of $k$ within ${10}^{-4}$ accuracy.