Hirota-Miwa difference equations

introduction

• how to identify the standard objects of quantum integrable systems (transfer matrices, Baxter's $Q$-operators, etc.) with elements of classical nonlinear integrable difference equations ($\tau$-functions, Baker-Akhiezer functions, etc.).

dictionary

• The functional relation for commuting quantum transfer matrices of quantum integrable models is shown to coincide with the classical Hirota bilinear difference equation.
• This equation is equivalent to the completely discretized classical 2D Toda lattice with open boundaries.
• Elliptic solutions of Hirota's equation give a complete set of eigenvalues of the quantum transfer matrices.
• The elliptic solutions relevant to the 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 the bilinear discrete Hirota equation are analysed and a new determinant formula for eigenvalues of the quantum transfer matrices is obtained.

Baxter $Q$-operator

• Eigenvalues of Baxter's $Q$-operator are solutions to the auxiliary linear problems for the classical Hirota equation.
• Difference equations for eigenvalues of the $Q$-operators which generalize Baxter's three-term $T-Q$-relation are derived

related items

• Hirota bilinear method
• T-system
• Determinant solutions of T-systems
• Octahedral recurrence
• Difference L-operators and TT-relations

expositions

• Zabrodin, A. 2012. “Bethe Ansatz and Hirota Equation in Integrable Models.” arXiv:1211.4428 [hep-Th, Physics:math-Ph] (November 19). http://arxiv.org/abs/1211.4428.
• Zabrodin, A. V. 1998. “Hirota Equation and Bethe Ansatz.” Theoretical and Mathematical Physics 116 (1) (July 1): 782–819. doi:10.1007/BF02557123.
• Wiegmann, P. 1997. “Bethe Ansatz and Classical Hirota Equation.” International Journal of Modern Physics B 11 (01n02) (January 20): 75–89. doi:10.1142/S0217979297000101.
• Zabrodin, A. V. 1997. “Hirota’s Difference Equations.” Theoretical and Mathematical Physics 113 (2) (November 1): 1347–1392. doi:10.1007/BF02634165. http://arxiv.org/abs/solv-int/9704001
• Zabrodin, A. V. 1997. “Bethe Ansatz and Classical Hirota Equations.” In Proceedings of the Second International A. D. Sakharov Conference on Physics (Moscow, 1996), 627–632. World Sci. Publ., River Edge, NJ. http://arxiv.org/abs/hep-th/9607162.

 

articles

• Krichever, I., O. Lipan, P. Wiegmann, and A. Zabrodin. 1997. “Quantum Integrable Models and Discrete Classical Hirota Equations.” Communications in Mathematical Physics 188 (2): 267–304. doi:10.1007/s002200050165. http://arxiv.org/abs/hep-th/9604080.
• T. Miwa, Proc. Japan. Acad. 58, 9 (1982).
• R. Hirota, Discrete analogue of a generalized Toda equation, J. Phys. Soc. Jpn. 50, 3785 (1981).