On the Equivalent Theory of the Generalized tau^{(2)}-model and the Chiral Potts Model with two Alternating Vertical Rapidities

arXiv:0710.2764v3 [cond-mat.stat-mech] 12 Feb 2008 On the Equivalent Theory of the Generalized τ (2)-model and the Chiral Potts Model with two Alternating Vertical Rapidities Shi-shyr Roan Institute of Mathematics Academia Sinica Taipei , Taiwan (email: maroan@gate.sinica.edu.tw ) Abstract By the Baxter’s Q72-operator method, we demonstrate the equivalent theory between the generalized τ (2)-model (other than two special cases with a pseudovacuum state) and the N- state chiral Potts model with two alternating vertical rapidities, where the degenerate models are included. As a consequence, the theory of the XXZ chain model associated to cyclic rep- resentations (with the parameter ς) of Uq(sl2) with qN = 1 for odd N is identified with either (for ςN = 1) the chiral Potts model with two superintegrable vertical rapidities, or (for ςN ̸= 1) the degenerate model for the selfdual solution of the star-triangle relation. In all these identi- fications, the transfer matrices T, bT of the chiral Potts model (including the degenerate ones) serve as the QR, QL-operators of the corresponding τ (2)-model, so that the functional relations hold as in the solvable N-state chiral Potts model. 2006 PACS: 05.50.+q, 03.65.Fd, 75.10.Pq 2000 MSC: 14H50, 39B72, 82B23 Key words: Generalized τ (2)-model, N-state chiral Potts model, Selfdual Potts model, Q-operator 1 1 Introduction In the study of N-state chiral Potts model (CPM) as a descendant of the six-vertex model, Bazhanov and Stroganov [14] found a five-parameter family of Yang-Baxter (YB) solutions for the asymmetric six-vertex R-matrix, which defines the generalized τ (2)-model, also known as the Baxter-Bazhanov- Stroganov model [7, 11, 14, 22]. The transfer chiral-Potts matrix arises as the Q-operator of the corresponding τ (2)-matrix [12, 14] by following the construction of Baxter’s Q-operator for the eight-vertex model in [6]. Hereafter in this paper, the CPM always means the ”checkerboard” type model with two vertical (alternating) rapidities as discussed in [12], where the functional-relation method was invented due to the lack of the ”difference” property of CPM rapidities in a high-genus curve. By counting the free parameters of CPM, one easily see that the τ (2)-models arisen from CPM form a three-parameter sub-family among all generalized τ (2)-models. The aim of this paper is to conduct the Q-operator investigation for an arbitrary generalized τ (2)-model along the line of Baxter’s Q72-operator in the eight-vertex model [4]. First, we note that a pseudovacuum state exists only for a certain special type of τ (2)-models, which can be studied by the powerful algebraic Bethe ansatz method [20, 23, 24, 35] as previously shown in [31]. Except those τ (2)-models possessing a pseudovacuum state, the main result of this paper can be loosely stated as ” the generalized τ (2)-models and CPM with two vertical rapidities are the equivalent theories provided degenerate versions of CPM are included”. The CPM transfer matrix will be derived as the Q-operator of the corresponding τ (2)-model in the functional-relation framework [12, 29]. Note that the τ (2)-model in this work is the trace of product of L-operator (2.12), which is invariant under gauge and scale transforms (2.21) (2.22). Using these transformations, one can always reduce the τ (2)-model to one in CPM with the alternating rapidities having the same temperature-like parameter k′. Hence the Q-operator is the CPM transfer matrix in [12], however the degenerate forms are necessarily included. Furthermore, Baxter extended the study of CPM transfer matrix and functional relations to some τ (2)-models [11] more general than those considered in this work. The generalized τ (2)- model of Baxter in [11] is an ”inhomogeneous” model of alternating rapidities with two k′s, not generally equal even by the gauge and scale transforms. Then the Boltzmann weights not necessarily satisfy the usual N-periodicity conditions, but replaced by a weaker condition ([11] (27)). By the observation that a special gauge transformation and the rescaling of spectral variables of the L-operator give rise to the equivalent τ (2)-models, a ”generic” τ (2)-model can be reduced to a τ (2)-model in CPM. Indeed, one can derive the quantitative description of the ”generic”-criterion about parameters in L-operator by the algebraic geometry study of these equivalent relations among τ (2)-models. As a consequence of this result, the conjectural boundary fusion relation [22, 31] holds for an arbitrary generalized τ (2)-model, hence the method of separation of variables can be applied in the study of τ (2)-models ([22] Theorem 2). Furthermore, the non-generic τ (2)-models are now the only remaining cases where an appropriate Q-operator is to be found in the theory. In this paper, we employ the Baxter’s techniques of producing Q72-operator of the root-of-unity eight-vertex model [4] to construct the QR, QL-, then Q-operator for a given τ (2)-model, as in the Q-o

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