First law and anisotropic Cardy formula for three-dimensional Lifshitz black holes

The aim of this paper is to confirm in new concrete examples that the semiclassical entropy of a three-dimensional Lifshitz black hole can be recovered through an anisotropic generalization of the Cardy formula derived from the growth of the number of states of a boundary nonrelativistic field theory. The role of the ground state in the bulk is played by the corresponding Lifshitz soliton obtained by a double Wick rotation. In order to achieve this task, we consider a scalar field nonminimally coupled to new massive gravity for which we study different classes of Lifshitz black holes as well as their respective solitons, including new solutions for a dynamical exponent $z=3$. The masses of the black holes and solitons are computed using the quasilocal formulation of conserved charges recently proposed by Gim et al. and based on the off-shell extension of the ADT formalism. We confirm the anisotropic Cardy formula for each of these examples, providing a stronger base for its general validity. Consistently, the first law of thermodynamics together with a Smarr formula are also verified.