The τ₂-model and the chiral Potts model revisited: completeness of Bethe equations from Sklyanin’s SOV method

Abstract

The most general cyclic representations of the quantum integrable τ 2 -model are analyzed. The complete characterization of the τ 2 -spectrum (eigenvalues and eigenstates) is achieved in the framework of Sklyanin's Separation of Variables (SOV) method by extending and adapting the ideas first introduced in [1, 2]: i) The determination of the τ 2 -spectrum is reduced to the classification of the solutions of a given functional equation in a class of polynomials. ii) The determination of the τ 2 -eigenstates is reduced to the classification of the solutions of an associated Baxter equation. These last solutions are proven to be polynomials for a quite general class of τ 2 -self-adjoint representations and the completeness of the associated Bethe ansatz type equations is derived. Finally, the following results are derived for the inhomogeneous chiral Potts model: i) Simplicity of the spectrum, for general representations. ii) Complete characterization of the chiral Potts spectrum (eigenvalues and eigenstates) and completeness of Bethe ansatz type equations, for the self-adjoint representations of τ 2 -model on the chiral Potts algebraic curves.

Antiperiodic dynamical 6-vertex model by separation of variables II: functional equations and form factors

“…It is therefore important to be able to pass from the discrete to the continuous picture or, in other words, to find an equivalent reformulation of the SOV discrete characterization of the transfer matrix spectrum and eigenstates in terms of some particular class of solutions on the whole complex plane C of a functional T -Q equation of Baxter's type. Note that the existence of such a reformulation has been already proven for several integrable quantum models solved by SOV [35,36,21], notably for the antiperiodic XXZ spin chain [42] which constitutes a limiting case of the present model (see Remark 2.2).…”

The open XXX spin chain in the SoV framework: scalar product of separate states

“…Open spin chains with non-diagonal boundary conditions have also been studied by means of the quantum Separation of Variables (SoV) [15,16,18,19,36,37]. This method, first introduced by Sklyanin in the QISM framework as an alternative to ABA for solving models in which a reference state cannot be identified [38], has recently shown to be applicable to a large class of models [39][40][41][42][43][44][45][46]. It has in particular permitted to construct the complete set of eigenstates for the spin chains with the most general (unconstrained) boundary terms [19,36,37].…”

Transfer matrix spectrum for cyclic representations of the 6-vertex reflection algebra I

“…Proof. The proof of this type of reformulation of the spectrum is by now quite standard and it has been proven for several models once they admit SoV description, see for example [36,[94][95][96][101][102][103]. So we will try to point out just some main features of the proof.…”

“…The parameters of the τ 2 -Lax operators are written in terms of the coordinate of the points p, r n , q n by using the equations (5.3) of the paper [96]. Then we have that the points r n , q n are elements of C k if and only if beyond (A.39) the parameters of the τ 2 -Lax operators satisfy the following conditions: α n β n = a n c n (A.40) and c 0 α n q 1/2 a n p + q 1/2 c 0 α n c n p = k 1 + c 2 0 α 2 n c n a n p .…”

“…In order to study the general representations and boundary conditions we have to go beyond traditional methods which do not apply for these general settings. This is done by developing the Sklyanin's SoV method [84][85][86][87] for this class of models, a method that has the advantage to lead (mainly by construction) to the complete characterization of the spectrum (eigenvalues and eigenvectors) and has proven to be applicable for a large variety of integrable quantum models [28][29][30][31][32][33][34][35][36][88][89][90][91][92][93][94][95][96][97][98][99][100][101][102], where traditional methods fail. Moreover, the SoV approach has the advantage to allow also for the study of the dynamics of the models as it leads to universal determinant formulae for matrix elements of local operators on transfer matrix eigenstates as shown for different classes of models, first in [97], and then in many other cases in [33,88,89,99,100,103].…”