Critical Casimir effect in a disordered $O(2)$-symmetric model

Critical Casimir effect appears when critical fluctuations of an order parameter interact with classical boundaries. We investigate this effect in the setting of a Landau-Ginzburg model with continuous symmetry in the presence of quenched disorder. The quenched free energy is written as an asymptotic series of moments of the models partition function. Our main result is that, in the presence of a strong disorder, Goldstone modes of the system contribute either with an attractive or with a repulsive force. This result was obtained using the distributional zeta-function method without relying on any particular ansatz in the functional space of the moments of the partition function.