Phantom BTZ black holes: Thermal properties under perturbative corrections

This study investigates the thermodynamics of phantom BTZ black holes by incorporating leading-order perturbative corrections arising from small statistical fluctuations around equilibrium. Starting from the phantom BTZ black holes review, we describe a modified action in three-dimensional spacetime that includes coupling with a Maxwell or phantom field. The analysis derives the corresponding field equations and obtains exact solutions for the metric and thermodynamic quantities. The corrected entropy is computed using the steepest descent method. It is expressed in terms of the leading-order entropy and Hawking temperature. Standard thermodynamic relations yield the corrected mass, Helmholtz free energy, specific heat, and Gibbs free energy. These corrections reveal significant deviations from classical results, particularly in the small black hole regime where statistical effects become prominent. Graphical analysis shows that the corrected entropy becomes negative for sufficiently small black holes, indicating potential limitations in thermodynamic stability under perturbative corrections. Furthermore, the influence of thermal fluctuations proves substantial for the small black holes and negligible for larger black holes. In this work, we have also calculated critical points and critical compressibility factor ($Z_c$) of the phantom BTZ black hole, treating it as a Van der Waals fluid. This work provides a comprehensive understanding of how statistical fluctuations modify the thermodynamics of phantom BTZ black holes. It underscores the necessity of including such corrections in realistic models of black hole thermodynamics in $(2+1)$ dimensions.