Discrete anchoring: Bulk-substrate coupling in lattice models

We formulate the anchoring problem for discrete-state lattice models. Anchoring is the selection of a bulk equilibrium state from a degenerate set of equivalent equilibrium states in semi-infinite samples in contact with a substrate, a phenomenon widely discussed in the context of liquid crystalline displays. As a concrete example we consider this problem for the three-state Potts model employing two different approximations, viz., a layered mean-field approximation and a Bethe lattice approach. The anchoring behaviour of the model is shown to be completely determined by the symmetry properties of the Hamiltonian.