On the Form Factors of Local Operators in the Bazhanov–Stroganov and Chiral Potts Models

Abstract

We consider general cyclic representations of the 6-vertex Yang-Baxter algebra and analyze the associated quantum integrable systems, the Bazhanov-Stroganov model and the corresponding chiral Potts model on finite size lattices. We first determine the propagator operator in terms of the chiral Potts transfer matrices and we compute the scalar product of separate states (including the transfer matrix eigenstates) as a single determinant formulae in the framework of Sklyanin's quantum separation of variables. Then, we solve the quantum inverse problem and reconstruct the local operators in terms of the separate variables. We also determine a basis of operators whose form factors are characterized by a single determinant formulae. This implies that the form factors of any local operator are expressed as finite sums of determinants. Among these form factors written in determinant form are in particular those which will reproduce the chiral Potts order parameters in the thermodynamic limit. The results presented here are the generalization to the present models associated to the most general cyclic representations of the 6-vertex Yang-Baxter algebra of those we derived for the lattice sine-Gordon model.

Antiperiodic spin-1/2 XXZ quantum chains by separation of variables: Complete spectrum and form factors

“…This program has been already realized for several integrable quantum models as we will summarize in the following. In the case of the cyclic representations of the 6-vertex Yang-Baxter algebra like the lattice sine-Gordon model and the τ 2 -model (of special interest for the connection with chiral Potts model) the form factors of local operators have been derived in [53] and [54], respectively. In [83] this approach is developed for the higher spin representations of highest weight type of the rational 6-vertex Yang-Baxter algebra which in the homogeneous limit leads to the higher spin XXX antiperiodic quantum chain.…”

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

“…One of our motivations to the present work is to define the required SOV setup to generalize the approach used in [82] to the τ 2 -model associated to the most general cyclic representations of the 6-vertex Yang-Baxter algebra, such a generalization will be presented in [121]. There, the obtained reconstruction of local operators in terms of quantum separate variables and the derived determinant formula for the scalar products of states are used to compute the form factors of local operators on the transfer matrix eigenstates and to express them as sums of determinants given by simple modifications of the scalar product ones.…”

On determinant representations of scalar products and form factors in the SoV approach: the XXX case

“…The key role in these representations is played by a Vandermonde determinant and its various (straightforward) dressing, making the computation of scalar products in the SoV framework much simpler and transparent compared to the one obtained in the algebraic Bethe ansatz settings. Such representations have quite universal character and properties as it has been shown in many following examples [44,[46][47][48]87,88]. Moreover it can be anticipated that such features will generalize to even more complicated systems associated to higher rank quantum groups, making the SoV approach even more appealing.…”