Generating function for scalar products in the algebraic Bethe ansatz
- PDF / 506,904 Bytes
- 11 Pages / 612 x 792 pts (letter) Page_size
- 90 Downloads / 199 Views
GENERATING FUNCTION FOR SCALAR PRODUCTS IN THE ALGEBRAIC BETHE ANSATZ N. A. Slavnov∗
We construct a family of determinant representations for scalar products of Bethe vectors in models with gl(3) symmetry. This family is defined by a single generating function containing arbitrary complex parameters but is independent of their specific values. Choosing these parameters in different ways, we can obtain different determinant representations.
Keywords: scalar product, generating function, determinant DOI: 10.1134/S004057792009010X
1. Introduction The quantum inverse scattering method (QISM) developed by the Leningrad school [1]–[3] allows finding the spectra of the Hamiltonians of quantum integrable models. This method is also used to calculate correlation functions. Compact representations for scalar products of Bethe vectors are needed for this. In turn, the algebraic Bethe ansatz (ABA) (part of the QISM) is used to solve the last problem. The problem of calculating scalar products was actively investigated immediately after the ABA method was created [4]–[7]. A determinant representation in a particular case of scalar products was found in [8]. In this specific case, one Bethe vector remains generic (off-shell), and the other is an on-shell Bethe vector (i.e., an eigenvector of the transfer matrix). The determinant representation for these scalar products opened a way to study form factors and correlation functions in models with the gl(2) symmetry and their q-deformations [9]–[14]. It also turned out to be useful for numerically analyzing correlation functions [15]– [18]. As shown in [19], the determinant representations for scalar products are a consequence of the formulas for the action of the transfer matrix on off-shell Bethe vectors. Using these formulas, we can easily obtain a system of linear equations for scalar products of Bethe vectors of which one is on-shell. Because any system of linear equations is solved in terms of determinants, the scalar products are also expressed in terms of determinants. But the abovementioned formula for the action of the transfer matrix on off-shell Bethe vectors is typical for models with a rank-1 symmetry. The action of the transfer matrix becomes more complex in models with a symmetry of higher rank. Therefore, it has not yet been possible to obtain a closed system of linear equations for scalar products in such models. Nevertheless, determinant representations for scalar products have recently been obtained for models with the gl(3) and gl(2|1) symmetries [20]–[22], but we ∗
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia, e-mail: [email protected].
This research was supported by the Russian Foundation for Basic Research (Grant No. 18-01-00273a). Prepared from an English manuscript submitted by the author; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 204, No. 3, pp. 453–465, September, 2020. Received January 22, 2020. Revised January 22, 2020. Accepted March 22, 2020. 1216
c 2020 Pleiades Publishing, Ltd. 0040-577
Data Loading...
