On a discrete version of the CP1 sigma model and surfaces immersed in R-3

We present a discretization of the CP1 sigma model. We show that the discrete CP1 sigma model is described by a nonlinear partial second-order difference equation with rational nonlinearity. To derive discrete surfaces immersed in three-dimensional Euclidean space a 'complex' lattice is introduced. The so-obtained surfaces are characterized in terms of the quadrilateral cross-ratio of four surface points. In this way we prove that all surfaces associated with the discrete CP1 sigma model are of constant mean curvature. An explicit example of such discrete surfaces is constructed.