I. Krichever, O. Lipan, P. Wiegmann, A. Zabrodin
Quantum Integrable Models and Discrete Classical Hirota Equations
The paper is published: Comm. Math. Phys. 188, No. 2 (1997) 267-304

MSC:

58F07 Completely integrable systems (including systems with an infinite number of degrees of freedom)

39A10 Difference equations, See also {33Dxx}

82B23 Exactly solvable models

Abstract: Functional relation for commuting quantum transfer matrices
of quantum integrable models is
identified with classical Hirota's
bilinear difference equation.
This equation is equivalent to the
completely discretized classical 2D Toda lattice with open boundaries.
The standard objects of quantum integrable models are identified
with elements of classical nonlinear integrable difference equation.
In particular, elliptic solutions of Hirota's equation give
complete set of eigenvalues of the quantum transfer matrices.
Eigenvalues of
Baxter's $Q$-operator are solutions to the
auxiliary linear problems for classical Hirota's equation.
The elliptic solutions relevant to Bethe ansatz are studied.
The nested Bethe ansatz equations for $A_{k-1}$-type models
appear as discrete time equations of motions for zeros of
classical $\tau$-functions and Baker-Akhiezer functions.
Determinant representations of the general solution
to bilinear discrete Hirota's equation and
a new determinant formula for eigenvalues of the
quantum transfer matrices
are obtained.