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Thermodynamics and Cardy-like formula for nonminimally dressed, charged Lifshitz black holes in New Massive Gravity

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