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)

Abstract

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
Volume35
Issue number5
Publication statusPublished - 2004
Externally publishedYes

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title = "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",
abstract = "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 quantumseparation of variables for 2 + 1-dimensional models)",
author = "Sergey Sergeev",
year = "2004",
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TY - JOUR

T1 - Quantum integrable models in discrete 2+1-dimensional space-time

T2 - Auxiliary linear problem on a lattice, zero-curvature representation, isospectral deformation of the Zamolodchikov-Bazhanov-Baxter model

AU - Sergeev, Sergey

PY - 2004

Y1 - 2004

N2 - 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 quantumseparation of variables for 2 + 1-dimensional models)

AB - 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 quantumseparation of variables for 2 + 1-dimensional models)

M3 - Article

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JF - Physics of Particles and Nuclei

SN - 0090-4759

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ER -