Functional equations and quantum separation of variables for 3d spin models

The recently proposed invariant formulation of the auxiliary linear problem for 3d integrable models provides several new ideas for solving the spectral problem of 3d spin models, e.g., the Zamolodchikov–Bazhanov–Baxter model in its vertex formulation. This paper announces results following from the invariant formulation. We formulate the class of 3d spin models that are essentially appropriately parameterized inhomogeneous Zamolodchikov–Bazhanov–Baxter models, present an expression for the generating function of the complete set of matrices commuting with the transfer matrix of this model (integrals of motion), give the functional equations defining the eigenvalues of the integrals of motion and the transfer matrices, explicitly describe the groupoid of isospectral transformations of the initial system of integrals of motion, and finally give an explicit parameterization of a projection operator onto the separated states in the sense of the quantum separation of variables (functional Bethe ansatz)