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arXiv:nlin/0209027 · CRC Press eBooks

Ladder Operators for Integrable One-Dimensional Lattice Models

M. C. Takizawa, Jon Links

Published Nov 30, 2003• 1 citation
arXivHTMLPDF

Abstract

A generalised ladder operator is used to construct the conserved operators for any one-dimensional lattice model derived from the Yang-Baxter equation. As an example, the low order conserved operators for the XYh model are calculated explicitly.
DOI
10.1201/9781482269079-75

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1 Yang-Baxter algebra and generation of quantum integrable models

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An operator defoemed quantum algebra is discovered exploiting the quantum Yang-Baxter equation with trigonometric R-matrix. This novel Hopf algebra along with its q → 1 limit seem to be the most general Yang-Baxter algebra underlying quantum integrable systems. Three different directions of application of this algebra in integrable systems depending on different sets of values of deforming operators are identified. Fixed values on the whole lattice yield subalgebras linked to standard quantum integrable models, while the associated Lax operators generate and classify them in an unified way. Variable values construct a new series of quantum integrable inhomogeneous models. Fixed but different values at different lattice sites can produce a novel class of integrable hybrid models including integrable matter-radiation models and quantum field models with defects, in particular, a new quantum integrable sine-Gordon model with defect. PACS numbers 02.30.Ik, 02.20.Uw, 05.50+q, 03.65.Fd 1

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The theory of constructing integrable systems using solutions to the Yang-Baxter equation (YBE) is well established. The conserved operators can be obtained by a series expansion for the family of commuting transfer matrices. Another approach is to use the ladder operator, which permits a recursive method through repeated commutators, to obtain the conserved operators. The ladder operator for models where the solution of the YBE has the difference property is well known. Recently it has been shown that a ladder operator exists for the Hubbard model, which does not have the difference property, and the present work extends this to a general theory for the construction of the ladder operator for any integrable system obtained through the YBE, with the XYh model as an example. The existence of ladder operators for integrable models constructed from other variants of the YBE, such as closed chains with quantum algebra symmetry and the Heisenberg model with single site impurity, is also considered.
The one-dimensional Hubbard model is integrable in the sense that it has an infinite family of conserved currents. We explicitly construct a ladder operator which can be used to iteratively generate all of the conserved current operators. This construction is different from that used for Lorentz invariant systems such as the Heisenberg model. The Hubbard model is not Lorentz invariant, due to the separation of spin and charge excitations. The ladder operator is obtained by a very general formalism which is applicable to any model that can be derived from a solution of the Yang-Baxter equation.
An operator defoemed quantum algebra is discovered exploiting the quantum Yang-Baxter equation with trigonometric R-matrix. This novel Hopf algebra along with its q → 1 limit seem to be the most general Yang-Baxter algebra underlying quantum integrable systems. Three different directions of application of this algebra in integrable systems depending on different sets of values of deforming operators are identified. Fixed values on the whole lattice yield subalgebras linked to standard quantum integrable models, while the associated Lax operators generate and classify them in an unified way. Variable values construct a new series of quantum integrable inhomogeneous models. Fixed but different values at different lattice sites can produce a novel class of integrable hybrid models including integrable matter-radiation models and quantum field models with defects, in particular, a new quantum integrable sine-Gordon model with defect. PACS numbers 02.30.Ik, 02.20.Uw, 05.50+q, 03.65.Fd 1