Geodesic
Generating function for scalar products in the algebraic Bethe ansatz
Teoretičeskaâ i matematičeskaâ fizika, Tome 204 (2020) no. 3, pp. 453-465 Cet article a éte moissonné depuis la source Math-Net.Ru

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We construct a family of determinant representations for scalar products of Bethe vectors in models with $\mathfrak{gl}(3)$ 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.
Mots-clés : scalar product
Keywords: generating function, determinant. 
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N. A. Slavnov. Generating function for scalar products in the algebraic Bethe ansatz. Teoretičeskaâ i matematičeskaâ fizika, Tome 204 (2020) no. 3, pp. 453-465. http://geodesic.mathdoc.fr/item/TMF_2020_204_3_a9/

[1] E. K. Sklyanin, L. A. Takhtadzhyan, L. D. Faddeev, “Kvantovyi metod obratnoi zadachi. I”, TMF, 40:2 (1979), 194–220 | DOI | MR

[2] L. A. Takhtadzhyan, L. D. Faddeev, “Kvantovyi metod obratnoi zadachi i $XYZ$ model Geizenberga”, UMN, 34:5(209) (1979), 13–63 | DOI | MR

[3] L. D. Faddeev, “How the algebraic Bethe ansatz works for integrable models”, Symmétries quantiques [Quantum Symmetries], Proceedings of the Les Houches summer school, Session LXIV (Les Houches, France, August 1 – September 8, 1995), eds. A. Connes, K. Gawedzki, J. Zinn-Justin, North-Holland, Amsterdam, 1998, 149–219 | MR | Zbl

[4] V. E. Korepin, “Calculation of norms of Bethe wave functions”, Commun. Math. Phys., 86:3 (1982), 391–418 | DOI | MR | Zbl

[5] A. G. Izergin, V. E. Korepin, “The quantum inverse scattering method approach to correlation functions”, Commun. Math. Phys., 94:1 (1984), 67–92 | DOI | MR | Zbl

[6] A. G. Izergin, “Statisticheskaya summa shestivershinnoi modeli v konechnom ob'eme”, Dokl. AN SSSR, 297:2 (1987), 331–333 | MR | Zbl

[7] V. E. Korepin, N. M. Bogoliubov, A. G. Izergin, Quantum Inverse Scattering Method and Correlation Functions, Cambridge Univ. Press, Cambridge, 1993 | MR

[8] N. A. Slavnov, “Vychislenie skalyarnykh proizvedenii volnovykh funktsii i formfaktorov v ramkakh algebraicheskogo anzatsa Bete”, TMF, 79:2 (1989), 232–240 | DOI | MR

[9] N. Kitanine, J. M. Maillet, V. Terras, “Correlation functions of the $XXZ$ Heisenberg spin-$1/2$ chain in a magnetic field”, Nucl. Phys. B, 567:3 (2000), 554–582, arXiv: math-ph/9907019 | DOI | MR | Zbl

[10] N. Kitanine, J. M. Maillet, N. A. Slavnov, V. Terras, “Spin-spin correlation functions of the $XXZ$-$1/2$ Heisenberg chain in a magnetic field”, Nucl. Phys. B, 641:3 (2002), 487–518, arXiv: hep-th/0201045 | DOI | MR | Zbl

[11] N. Kitanine, K. K. Kozlowski, J. M. Maillet, N. A. Slavnov, V. Terras, “Algebraic Bethe ansatz approach to the asymptotic behavior of correlation functions”, J. Stat. Mech., 2009:4 (2009), P04003, 66 pp., arXiv: 0808.0227 | DOI | MR

[12] F. Göhmann, A. Klümper, A. Seel, “Integral representations for correlation functions of the $XXZ$ chain at finite temperature”, J. Phys. A: Math. Gen., 37:31 (2004), 7625–7652, arXiv: hep-th/0405089 | DOI | MR

[13] F. Göhmann, A. Klümper, A. Seel, “Integral representation of the density matrix of the $XXZ$ chain at finite temperatures”, J. Phys. A: Math. Gen., 38:9 (2005), 1833–1842, arXiv: cond-mat/0412062 | DOI

[14] A. Seel, T. Bhattacharyya, F. Göhmann, A. Klümper, “A note on the spin-$1/2$ $XXZ$ chain concerning its relation to the Bose gas”, J. Stat. Mech., 2007:8 (2007), P08030, 10 pp., arXiv: 0705.3569 | DOI | MR

[15] J. S. Caux, J. M. Maillet, “Computation of dynamical correlation functions of Heisenberg chains in a magnetic field”, Phys. Rev. Lett., 95:7 (2005), 077201, 3 pp., arXiv: cond-mat/0502365 | DOI

[16] R. G. Pereira, J. Sirker, J. S. Caux, R. Hagemans, J. M. Maillet, S. R. White, I. Affleck, “Dynamical spin structure factor for the anisotropic spin-$1/2$ Heisenberg chain”, Phys. Rev. Lett., 96:25 (2006), 257202, 4 pp., arXiv: cond-mat/0603681 | DOI

[17] R. G. Pereira, J. Sirker, J. S. Caux, R. Hagemans, J. M. Maillet, S. R. White, I. Affleck, “Dynamical structure factor at small $q$ for the $XXZ$ spin-$1/2$ chain”, J. Stat. Mech., 2007:8 (2007), P08022, 64 pp., arXiv: 0706.4327 | DOI | MR

[18] J. S. Caux, P. Calabrese, N. A. Slavnov, “One-particle dynamical correlations in the one-dimensional Bose gas”, J. Stat. Mech., 2007:1 (2007), P01008, 21 pp., arXiv: cond-mat/0611321 | DOI

[19] S. Belliard, N. A. Slavnov, “Why scalar products in the algebraic Bethe ansatz have determinant representation”, JHEP, 10 (2019), 103, 17 pp., arXiv: 1908.00032 | DOI

[20] S. Belliard, S. Pakuliak, E. Ragoucy, N. A. Slavnov, “The algebraic Bethe ansatz for scalar products in $SU(3)$-invariant integrable models”, J. Stat. Mech., 2012:10 (2012), P10017, 25 pp., arXiv: 1207.0956 | DOI | MR

[21] N. A. Slavnov, “Scalar products in $GL(3)$-based models with trigonometric $R$-matrix. Determinant representation”, J. Stat. Mech., 2015:3 (2015), P03019, 25 pp., arXiv: 1501.06253 | DOI | MR

[22] A. Hutsalyuk, A. Lyashik, S. Pakuliak, E. Ragoucy, N. A. Slavnov, “Scalar products of Bethe vectors in models with $\mathfrak{gl}(2|1)$ symmetry 2. Determinant representation”, J. Phys. A, 50:3 (2017), 034004, 22 pp., arXiv: 1605.09189 | DOI | MR

[23] B. Pozsgay, W.-V. van G. Oei, M. Kormos, “On form factors in nested Bethe Ansatz systems”, J. Phys. A: Math. Gen., 45:46, 465007, 34 pp., arXiv: 1204.4037 | DOI | MR

[24] S. Belliard, S. Pakuliak, E. Ragoucy, N. A. Slavnov, “Form factors in $SU(3)$-invariant integrable models”, J. Stat. Mech., 2013:4 (2013), P04033, 16 pp., arXiv: 1211.3968 | DOI | MR

[25] S. Pakuliak, E. Ragoucy, N. A. Slavnov, “Form factors in quantum integrable models with $GL(3)$-invariant $R$-matrix”, Nucl. Phys. B, 881 (2014), 343–368, arXiv: 1312.1488 | DOI | MR

[26] S. Pakuliak, E. Ragoucy, N. A. Slavnov, “Zero modes method and form factors in quantum integrable models”, Nucl. Phys. B, 893 (2015), 459–481, arXiv: 1412.6037 | DOI | MR

[27] S. Pakuliak, E. Ragoucy, N. A. Slavnov, “${\rm GL}(3)$-based quantum integrable composite models. II. Form factors of local operators”, SIGMA, 11 (2015), 064, 18 pp., arXiv: 1502.01966 | MR

[28] A. Hustalyuk, A. Liashyk, S. Z. Pakulyak, E. Ragoucy, N. A. Slavnov, “Form factors of the monodromy matrix entries in $\mathfrak{gl}(2|1)$-invariant integrable models”, Nucl. Phys. B, 911 (2016), 902–927, arXiv: 1607.04978 | DOI

[29] J. Fuksa, N. A. Slavnov, “Form factors of local operators in supersymmetric quantum integrable models”, J. Stat. Mech., 2017:4 (2017), 043106, 21 pp., arXiv: 1701.05866 | DOI | MR

[30] A. N. Kirillov, F. A. Smirnov, “Reshenie nekotorykh kombinatornykh problem, voznikayuschikh pri vychislenii korrelyatorov v tochno-reshaemykh modelyakh”, Zap. nauchn. sem. LOMI, 164 (1987), 67–79 | Zbl

[31] S. Belliard, S. Pakuliak, E. Ragoucy, N. A. Slavnov, “Bethe vectors of $GL(3)$-invariant integrable models”, J. Stat. Mech., 2013:2 (2013), P02020, 24 pp. | DOI | MR