Classical tau-function for quantum spin chains

Alexandrov Alexander (1, 2) , Vladimir Kazakov (3, 4) , Sébastien Leurent (4) , Zengo Tsuboi (5, 6) , Anton Zabrodin (7, 2)

1 Mathematisches Institut [Freiburg]
2 ITEP - Institute of Theoretical and Experimental Physics [Moscow]
3 UPMC - Université Pierre et Marie Curie - Paris 6
4 LPTENS - Laboratoire de Physique Théorique de l'ENS [École Normale Supérieure]
5 Institut für Mathematik [Humboldt]
6 Institut fur Physik, Humboldt Universitat zu Berlin
7 NRU HSE, faculty of mathematics

For an arbitrary generalized quantum integrable spin chain we introduce a "master T -operator" which represents a generating function for commuting quantum transfer matrices constructed by means of the fusion procedure in the auxiliary space. We show that the functional relations for the transfer matrices are equivalent to an infinite set of model-independent bilinear equations of the Hirota form for the master T -operator, which allows one to identify it with $\tau$-function of an integrable hierarchy of classical soliton equations. In this paper we consider spin chains with rational GL(N)-invariant R-matrices but the result is independent of a particular functional form of the transfer matrices and directly applies to quantum integrable models with more general (trigonometric and elliptic) R-matrices and to supersymmetric spin chains.

Domaines: Physique mathématique [math-ph]

Alexandrov Alexander, Vladimir Kazakov, Sébastien Leurent, Zengo Tsuboi, Anton Zabrodin. Classical tau-function for quantum spin chains. Journal of High Energy Physics, 2013, 1309, pp.064. ⟨10.1007/JHEP09(2013)064⟩. ⟨hal-01325649⟩