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

Sergey Sergeev

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6 Citations (Scopus)


An invariant approach to quantum integrable models in wholly discrete 2 + 1-dimensional space–time 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 eigenstates of the models under consideration; the method is based on isospectral deformations (the method of quantum
separation of variables for 2 + 1-dimensional models)
Original languageEnglish
Pages (from-to)1-31
Number of pages61
JournalPhysics of Particles and Nuclei
Issue number5
Publication statusPublished - 2004
Externally publishedYes


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