Excited Kerr black holes with scalar hair

In the context of a complex scalar field coupled to Einstein gravity theory, we present a novel family of solutions of Kerr black holes with excited state scalar hair inspired by the work of Herdeiro and Radu in [Phys. Rev. Lett. 112, 221101 (2014)], which can be regarded as numerical solutions of rotating compact objects with excited scalar hair, including boson stars and black holes. In contrast to Kerr black holes with ground state scalar hair, we find that the first excited Kerr black holes with scalar hair have two types of nodes, including radial n(r) = 1 and angular n(theta) = 1 nodes. Moreover, in the case of both nodes the curves of the mass versus the frequency form nontrivial loops. Furthermore, we also show the numerical results of the second excited states with even parity, and find that the curves can be divided into two kinds: closed and open loops. We also study the dependence of the horizon area on angular momentum and Hawking temperature in these excited states.