Thermodynamical topology of quantum BTZ black hole

Among the study of black hole thermodynamics, topology offers a novel approach and perspective for classifying black hole systems. In this work, we explore the thermodynamical topology of the quantum Bañados-Teitelboim-Zanelli (BTZ) black hole by employing the concept of the generalized free energy. To fully characterize the thermodynamics, we introduce two distinct topological numbers. The first one is determined by an expression, denoted by $z$, derived from the free energy. Although it can provide us with some local physical explanations, sufficient physical significance still lacks from a global perspective. On the other hand, the second topological number is based on the entropy expression of the generalized free energy, leading to a more meaningful interpretation of its physical implications. This result highlights the natural choice of entropy as the domain variable for the generalized free energy. Regarding the second topological number, our analysis reveals a topological transition that is associated with the thermodynamical stability of the "cold" black hole state of the quantum BTZ black hole. And the thermodynamical topology of the BTZ black hole and quantum BTZ black hole can be different, which implies a significant impact of quantum effects on thermodynamics. Furthermore, our study suggests the existence of topological numbers beyond the conventional values of $\pm 1$, 0.