Photon orbits and phase transition for Letelier AdS black holes immersed in perfect fluid dark matter

We obtain an exact solution for spherically symmetric Letelier AdS black holes immersed in perfect fluid dark matter (PFDM). Considering the cosmological constant as the positive pressure of the system and volume as its conjugate variable, we analyze the thermodynamics of our black holes in the extended phase space. Owing to the background clouds of strings parameter (a) and the parameter endowed with PFDM (beta), we analyze the Hawking temperature, entropy, and specific heat. Furthermore, we investigate the relationship between the photon sphere radius and phase transition for the Letelier AdS black holes immersed in PFDM. Through the analysis, with a particular condition, non-monotonic behaviors are found between the photon sphere radius, impact parameter, PFDM parameter, temperature, and pressure. We can regard the changes in both the photon sphere radius and impact parameter before and after phase transition as the order parameter; their critical exponents near the critical point are equal to the same value, 1/2, similar to that in ordinary thermal systems. This indicates that a universal relation of gravity may exist near the critical point for a black hole thermodynamic system.