Holographic Entanglement Entropy of Anisotropic Minimal Surfaces in LLM Geometries

We calculate the holographic entanglement entropy (HEE) of the $mathbb{Z}_k$ orbifold of Lin-Lunin-Maldacena (LLM) geometries which are dual to the vacua of the mass-deformed ABJM theory with Chern-Simons level $k$. By solving the partial differential equations analytically, we obtain the HEEs for all LLM solutions with arbitrary M2 charge and $k$ up to $mu_0^2$-order where $mu_0$ is the mass parameter. The renormalized entanglement entropies are all monotonically decreasing near the UV fixed point in accordance with the $F$-theorem. Except the multiplication factor and to all orders in $mu_0$, they are independent of the overall scaling of Young diagrams which characterize LLM geometries. Therefore we can classify the HEEs of LLM geometries with $mathbb{Z}_k$ orbifold in terms of the shape of Young diagrams modulo overall size. HEE of each family is a pure number independent of the t Hooft coupling constant except the overall multiplication factor. We extend our analysis to obtain HEE analytically to $mu_0^4$-order for the symmetric droplet case.