734 research outputs found
Classical R-Operators and Integrable Generalizations of Thirring Equations
We construct different integrable generalizations of the massive Thirring
equations corresponding loop algebras in
different gradings and associated ''triangular'' -operators. We consider the
most interesting cases connected with the Coxeter automorphisms, second order
automorphisms and with ''Kostant-Adler-Symes'' -operators. We recover a
known matrix generalization of the complex Thirring equations as a partial case
of our construction.Comment: This is a contribution to the Proc. of the Seventh International
Conference ''Symmetry in Nonlinear Mathematical Physics'' (June 24-30, 2007,
Kyiv, Ukraine), published in SIGMA (Symmetry, Integrability and Geometry:
Methods and Applications) at http://www.emis.de/journals/SIGMA
Twisted rational r-matrices and algebraic Bethe ansatz: Application to generalized Gaudin and Richardson models
In the present paper we develop the algebraic Bethe ansatz approach to the case of non-skew-symmetric gl(2) circle times gl(2)-valued Cartan-non-invariant classical r-matrices with spectral parameters. We consider the two families of these r-matrices, namely, the two non-standard rational r-matrices twisted with the help of second order automorphisms and realize the algebraic Bethe ansatz method for them. We study physically important examples of the Gaudin-type and BCS-type systems associated with these r-matrices and obtain explicitly the Bethe vectors and the spectrum for the corresponding quantum hamiltonians in terms of solutions of Bethe equations. (C) 2021 The Author(s). Published by Elsevier B.V.0117U000240info:eu-repo/semantics/publishedVersio
"Doubled" generalized Landau-Lifshiz hierarchies and special quasigraded Lie algebras
Using special quasigraded Lie algebras we obtain new hierarchies of
integrable nonlinear vector equations admitting zero-curvature representations.
Among them the most interesting is extension of the generalized Landau-Lifshitz
hierarchy which we call "doubled" generalized Landau-Lifshiz hierarchy. This
hierarchy can be also interpreted as an anisotropic vector generalization of
"modified" Sine-Gordon hierarchy or as a very special vector generalization of
so(3) anisotropic chiral field hierarchy.Comment: 16 pages, no figures, submitted to Journal of Physics
The Generalized Lipkin-Meshkov-Glick Model and the Modified Algebraic Bethe Ansatz
We show that the Lipkin-Meshkov-Glick -fermion model is a particular case
of one-spin Gaudin-type model in an external magnetic field corresponding to a
limiting case of non-skew-symmetric elliptic -matrix and to an external
magnetic field directed along one axis. We propose an exactly-solvable
generalization of the Lipkin-Meshkov-Glick fermion model based on the
Gaudin-type model corresponding to the same -matrix but arbitrary external
magnetic field. This model coincides with the quantization of the classical
Zhukovsky-Volterra gyrostat. We diagonalize the corresponding quantum
Hamiltonian by means of the modified algebraic Bethe ansatz. We explicitly
solve the corresponding Bethe-type equations for the case of small fermion
number
“Generalized” algebraic Bethe ansatz, Gaudin-type models and Z p -graded classical r -matrices
AbstractWe consider quantum integrable systems associated with reductive Lie algebra gl(n) and Cartan-invariant non-skew-symmetric classical r-matrices. We show that under certain restrictions on the form of classical r-matrices “nested” or “hierarchical” Bethe ansatz usually based on a chain of subalgebras gl(n)⊃gl(n−1)⊃...⊃gl(1) is generalized onto the other chains or “hierarchies” of subalgebras. We show that among the r-matrices satisfying such the restrictions there are “twisted” or Zp-graded non-skew-symmetric classical r-matrices. We consider in detail example of the generalized Gaudin models with and without external magnetic field associated with Zp-graded non-skew-symmetric classical r-matrices and find the spectrum of the corresponding Gaudin-type hamiltonians using nested Bethe ansatz scheme and a chain of subalgebras gl(n)⊃gl(n−n1)⊃gl(n−n1−n2)⊃gl(n−(n1+...+np−1)), where n1+n2+...+np=n
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