Casimir effect of a rough membrane in 2 + 1 Hořava–Lifshitz theory

Abstract

Abstract We investigate the Casimir effect of a rough membrane within the framework of the Hořava–Lifshitz theory in $$2+1$$ 2 + 1 dimensions. Quantum fluctuations are induced by an anisotropic scalar field subject to Dirichlet boundary conditions. We implement a coordinate transformation to render the membrane completely flat, treating the remaining terms associated with roughness as a potential. The spectrum is obtained through perturbation theory and regularized using the $$\zeta $$ ζ -function method. We present an explicit example of a membrane with periodic border. Additionally, we consider the effect of temperature. Our findings reveal that the Casimir energy and force depend on roughness, the anisotropic scaling factor and temperature.