Quantum integrable models in discrete 2 + 1-dimensional space-time: Auxiliary linear problem on a lattice, zero-curvature representation, isospectral deformation of the Zamolodchikov-Bazhanov-Baxter model

An invariant approach to quantum integrable models in wholly discrete 2 + 1-dimensional spacetime is considered. An auxiliary linear problem on two-dimensional lattices generalizing quantum chains is formulated. A method of constructing a complete set of integrals of motion is given. For the two-dimensional lattices, we formulate and solve a zero-curvature representation allowing us to construct integrable evolutionary mappings. We place special emphasis on finite-dimensional representations of the algebra of observables, which exist if the Weyl algebra parameter is at a rational point of a unit circle (so-called "root of unity"). For this case, we derive a universal functional eigenvalue equation for integrals of motion. A groupoid of isospectral deformations is constructed for the finite-dimensional representations of the algebra of observables. Because the systems under consideration are finite-dimensional at the root of unity, the integrable systems can be treated as models of statistical mechanics on three-dimensional lattices. We formulate a method of constructing eigen-states of the models under consideration; the method is based on isospectral deformations (the method of quantum separation of variables for 2 + 1-dimensional models).