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arXiv:hep-th/9610039 · International Journal of Modern Physics B

Discrete Hirota's Equation in Quantum Integrable Models

A. Zabrodin

Published Oct 30, 1997• 36 citations
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Abstract

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.

DOI
10.1142/s0217979297001520

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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.
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A brief non-technical review of the recent study of classical integrable structures in quantum integrable systems is given. It is explained how to identify the standard objects of quantum integrable systems (transfer matrices, Baxter's QQQ-operators, etc) with elements of classical non-linear integrable difference equations (τττ-functions, Baker-Akhiezer functions, etc). The nested Bethe ansatz equations for Ak−1A_{k-1}Ak−1​-type models emerge as discrete time equations of motion for zeros of classical τττ-functions and Baker-Akhiezer functions. The connection with discrete time Ruijsenaars-Schneider system of particles is discussed.
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