The aim of this letter 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 non-relativistic 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.
In arbitrary dimension, we consider a theory described by the most general quadratic curvature corrections of Einstein gravity together with a self-interacting nonminimally coupled scalar field. This theory is shown to admit five different families of Lifshitz black holes dressed with a nontrivial scalar field. The entropy of these configurations is microscopically computed by means of a higherdimensional anisotropic Cardy-like formula where the role of the ground state is played by the soliton obtained through a double analytic continuation. This involves calculating the correct expressions for the masses of the higher-dimensional Lifshitz black hole as well as their corresponding soliton. The robustness of this Cardy-like formula is checked by showing that the microscopic entropy is in perfect agreement with the gravitational Wald entropy. Consequently, the calculated global charges are compatible with the first law of thermodynamics as well as an anisotropic higher-dimensional version of the Smarr formula. Some of these configurations exist on Lifshitz critical points of the theory where all their extensive thermodynamic quantities vanish.
In three dimensions, we consider the Einstein-Maxwell Lagrangian dressed by a nonminimally coupled scalar field in New Massive Gravity. For this theory, we provide two families of electrically charged Lifshitz black holes where their metric functions depend only on an integration constant. We calculate their masses using the quasilocal approach, as well as their entropy and electric charge. These charged configurations are interpreted as extremal in the sense that the mass vanishes identically while the entropy and electric charge are non zero thermodynamic quantities. Using these examples, we corroborate that the semiclassical entropy can be recovered through a charged Cardy-like formula, involving the corresponding magnetically charged solitons obtained by a double Wick rotation. Finally, the first law of thermodynamics, as well as the Smarr formula are also verified.
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