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arXiv:1804.01749 · SciPost Physics

Solution of Baxter equation for the $q$-Toda and Toda$_2$ chains by NLIE

O. Babelon, Karol Kajetan Kozlowski, Vincent Pasquier

Published Oct 18, 2018• 5 citations
arXivHTMLPDF

Abstract

We construct a basis of solutions of the scalar t-Q equation describing the spectrum of the q-Toda and Toda _2 2 chains by using auxiliary non-linear integral equations. Our construction allows us to provide quantisation conditions for the spectra of these models in the form of thermodynamic Bethe Ansatz-like equations.
DOI
10.21468/scipostphys.5.4.035

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Solution of Baxter equation for the qqq-Toda and Toda2_22​ chains by NLIE
A review of the classical and quantum thermodynamic properties of the Toda chain is provided, together with a survey of the techniques that have been used to work them out, i.e., the bilateral Laplace transform method for the classical system, the Bethe–Ansatz and the variational path-integral approach for the quantum one. For the classical Toda chain we also recall some of the main dynamical features, i.e. the integrability of the model and the soliton-like solutions of the equations of motion. In the quantum case a comparison between the Bethe–Ansatz and the variational method is made. In particular it is shown as the latter offers the possibility of also evaluating static correlation functions.
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
The recent progress in revealing classical integrable structures in quantum models solved by Bethe ansatz is reviewed. Fusion relations for eigenvalues of quantum transfer matrices can be written in the form of classical Hirota's bilinear difference equation. This equation is also known as the completely discretized version of the 2D Toda lattice. We explain how one obtains the specific quantum results by solving the classical equation. The auxiliary linear problem for the Hirota equation is shown to generalize Baxter's T-Q relation.