Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:nlin/0009027

Relativistic Toda chain

G. P. Pronko, С. М. Сергеев

Published Sep 14, 2000• 7 citations
arXivHTMLPDF

Abstract

Investigated is the relativistic periodic Toda chain, to each site of which the ultra-local Weyl algebra is associated. Weyl's $q$ we are considering here is restricted to be inside the unit circle. Quantum Lax operators of the model are intertwined by six vertex $R$-matrix. Both independent Baxter's $Q$-operators are constructed explicitly as seria over local Weyl generators. The operator-valued wronskian of $Q$-s is also calculated.

DOI
10.48550/arxiv.nlin/0009027

Related papers

Quantum relativistic Toda chain

G. P. Pronko, С. М. Сергеев

Investigated is the quantum relativistic periodic Toda chain, to each site of which the ultra‐local Weyl algebra is associated. Weyl’s q we are considering here is restricted to be inside the unit circle. Quantum Lax operators of the model are intertwined by six‐vertex R‐matrix. Both independent Baxter’s Q‐operators are constructed explicitly as seria over local Weyl generators. The operator‐valued Wronskian of R‐matrix. Both independent Baxter’s Qs is also calculated.

Journal of Applied Mathematics•Jan 1, 2001•9 citations

Generation of a quantum integrable class of discrete-time or relativistic periodic Toda chains

Kundu, A

A new integrable class of quantum models representing a family of different discrete-time or relativistic generalisations of the periodic Toda chain (TC), including that of a recently proposed classical close to TC model [7] is presented. All such models are shown to be obtainable from a single ancestor model at different realisations of the underlying quantised algebra. As a consequence the $(2\\times 2)$ Lax operators and the associated quantum $R$-matrices for these models are easily derived ensuring their quantum integrability. It is shown that the functional Bethe ansatz developed for the quntum TC is trivially generalised to achieve seperation of variables also for the present models.

•Jan 1, 1994•17 citations

Baxter Operator and Baxter Equation for q-Toda and Toda2 Chains

O. Babelon, Karol Kajetan Kozlowski, Vincent Pasquier

We construct the Baxter operator [Formula: see text] for the [Formula: see text]-Toda chain and the Toda2 chain (the Toda chain in the second Hamiltonian structure). Our construction builds on the relation between the Baxter operator and Bäcklund transformations that were unravelled in [13]. We construct a number of quantum intertwiners ensuring the commutativity of [Formula: see text] with the transfer matrix of the models and of the [Formula: see text]’s between each other. Most importantly, [Formula: see text] is modular invariant in the sense of Faddeev. We derive the Baxter equation for the eigenvalues [Formula: see text] of [Formula: see text] and show that these are entire functions of [Formula: see text]. This last property will ultimately lead to the quantization of the spectrum for the considered Toda chains, in a subsequent publication [1]. This work is dedicated to the memory of L. D. Faddeev

arXiv:1803.06196•Reviews in Mathematical Physics•Jun 5, 2018•14 citations

Relativistic Toda chain at root of unity II. Modified Q-operator

Stanislav Zdislavovich Pakuliak, С. М. Сергеев

Matrix elements of quantum intertwiner as well as the modified Q-operator for the quantum relativistic Toda chain at root of unity are constructed explicitly. Modified Q-operators make isospectrality transformations of quantum transfer matrices so that the classical counterparts of Q-operators correspond to the Baecklund isospectrality transformations of the classical transfer matrices. Separated vectors for the Functional Bethe Ansatz are constructed with the help of modified Q-operators.

arXiv:nlin/0107062•Jul 27, 2001•5 citations

Baxterization ofGLq(2) and its application to the Liouville model and some other models on a lattice

Andrei Bytsko

We develop the Baxterization approach to (an extension of) the quantum group GL q (2). We introduce two matrices which play the role of spectral parameter-dependent L -matrices and observe that they are naturally related to two different comultiplications. Using these comultiplication structures, we find the related fundamental R-operators in terms of powers of coproducts and also give their equivalent forms in terms of quantum dilogarithms. The corresponding quantum local Hamiltonians are given in terms of logarithms of positive operators. An analogous construction is developed for the q -oscillator and Weyl algebras using the fact that their algebraic and coalgebraic structures can be obtained as reductions of those for the quantum group. As an application, the lattice Liouville model, the q -DST model, the Volterra model, a lattice regularization of the free field and the relativistic Toda model are considered.

arXiv:0712.3301•Journal of Physics A Mathematical and Theoretical•Apr 29, 2008•1 citation

1 Yang-Baxter algebra and generation of quantum integrable models

Anjan Kundu

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

•Jan 1, 2008•6 citations
Investigated is the relativistic periodic Toda chain, to each site of which the ultra-local Weyl algebra is associated. Weyl's qqq we are considering here is restricted to be inside the unit circle. Quantum Lax operators of the model are intertwined by six vertex RRR-matrix. Both independent Baxter's QQQ-operators are constructed explicitly as seria over local Weyl generators. The operator-valued wronskian of QQQ-s is also calculated.
A new integrable class of quantum models representing a family of different discrete-time or relativistic generalisations of the periodic Toda chain (TC), including that of a recently proposed classical close to TC model [7] is presented. All such models are shown to be obtainable from a single ancestor model at different realisations of the underlying quantised algebra. As a consequence the (2times2)(2\\times 2)(2times2) Lax operators and the associated quantum RRR-matrices for these models are easily derived ensuring their quantum integrability. It is shown that the functional Bethe ansatz developed for the quntum TC is trivially generalised to achieve seperation of variables also for the present models.
We construct the Baxter operator [Formula: see text] for the [Formula: see text]-Toda chain and the Toda2 chain (the Toda chain in the second Hamiltonian structure). Our construction builds on the relation between the Baxter operator and Bäcklund transformations that were unravelled in [13]. We construct a number of quantum intertwiners ensuring the commutativity of [Formula: see text] with the transfer matrix of the models and of the [Formula: see text]’s between each other. Most importantly, [Formula: see text] is modular invariant in the sense of Faddeev. We derive the Baxter equation for the eigenvalues [Formula: see text] of [Formula: see text] and show that these are entire functions of [Formula: see text]. This last property will ultimately lead to the quantization of the spectrum for the considered Toda chains, in a subsequent publication [1]. This work is dedicated to the memory of L. D. Faddeev
We develop the Baxterization approach to (an extension of) the quantum group GL q (2). We introduce two matrices which play the role of spectral parameter-dependent L -matrices and observe that they are naturally related to two different comultiplications. Using these comultiplication structures, we find the related fundamental R-operators in terms of powers of coproducts and also give their equivalent forms in terms of quantum dilogarithms. The corresponding quantum local Hamiltonians are given in terms of logarithms of positive operators. An analogous construction is developed for the q -oscillator and Weyl algebras using the fact that their algebraic and coalgebraic structures can be obtained as reductions of those for the quantum group. As an application, the lattice Liouville model, the q -DST model, the Volterra model, a lattice regularization of the free field and the relativistic Toda model are considered.
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