Casimir-Lifshitz torque between spatially anisotropic materials: A quantum field theory investigation

We present a theoretical investigation of the Casimir-Lifshitz torque between black phosphorus and the liquid crystal material 4-cyano-4-n-pentylcyclohexane-phenyl. Employing quantum field theory methods, we derive an effective action that describes the optical properties of black phosphorus, based on a low-energy effective Hamiltonian. We obtain an analytical expression for the Fresnel matrix of the black-phosphorus-vacuum system by modifying Maxwell's equations. Our analysis reveals that the wavelength-dependent permittivity function plays a crucial role in the study of Casimir-Lifshitz torque in two-dimensional systems.