Scalar products of Bethe vectors in the generalized algebraic
Teoretičeskaâ i matematičeskaâ fizika, Tome 217 (2023) no. 1, pp. 179-203
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We consider an $XYZ$ spin chain within the framework of the generalized algebraic Bethe ansatz. We study scalar products of the transfer matrix eigenvectors and arbitrary Bethe vectors. In the particular case of free fermions, we obtain explicit expressions for the scalar products with different number of parameters in two Bethe vectors.
Keywords: generalized algebraic Bethe ansatz, Bethe vectors, gauge transformed monodromy matrix, scalar products.
@article{TMF_2023_217_1_a9,
     author = {G. Kulkarni and N. A. Slavnov},
     title = {Scalar products of {Bethe} vectors in the~generalized algebraic},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {179--203},
     year = {2023},
     volume = {217},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2023_217_1_a9/}
}
TY  - JOUR
AU  - G. Kulkarni
AU  - N. A. Slavnov
TI  - Scalar products of Bethe vectors in the generalized algebraic
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2023
SP  - 179
EP  - 203
VL  - 217
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/TMF_2023_217_1_a9/
LA  - ru
ID  - TMF_2023_217_1_a9
ER  - 
%0 Journal Article
%A G. Kulkarni
%A N. A. Slavnov
%T Scalar products of Bethe vectors in the generalized algebraic
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2023
%P 179-203
%V 217
%N 1
%U http://geodesic.mathdoc.fr/item/TMF_2023_217_1_a9/
%G ru
%F TMF_2023_217_1_a9
G. Kulkarni; N. A. Slavnov. Scalar products of Bethe vectors in the generalized algebraic. Teoretičeskaâ i matematičeskaâ fizika, Tome 217 (2023) no. 1, pp. 179-203. http://geodesic.mathdoc.fr/item/TMF_2023_217_1_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, arXiv: hep-th/9605187 | MR | Zbl

[4] 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

[5] V. E. Korepin, “Dual field formulation of quantum integrable models”, Commun. Math. Phys., 113:2 (1987), 177–190 | DOI | MR

[6] T. Kojima, V. E. Korepin, N. A. Slavnov, “Determinant representation for dynamical correlation function of the quantum nonlinear Schrödinger equation”, Commun. Math. Phys., 188:3 (1997), 657–689, arXiv: hep-th/9611216 | DOI | MR

[7] M. Jimbo, K. Miki, T. Miwa, A. Nakayashiki, “Correlation functions of the $XXZ$ model for $\Delta-1$”, Phys. Lett. A, 168:4 (1992), 256–263, arXiv: hep-th/9205055 | DOI | MR

[8] 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

[9] 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

[10] N. Kitanine, J. M. Maillet, N. A. Slavnov, V. Terras, “Master equation for spin-spin correlation functions of the $XXZ$ chain”, Nucl. Phys. B, 712:3 (2005), 600–622, arXiv: ; 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: ; “A form factor approach to the asymptotic behavior of correlation functions”, 2011:12 (2011), P12010, 28 pp., arXiv: ; “Form factor approach to dynamical correlation functions in critical models”, 2012:9 (2012), P09001, 33 pp., arXiv: hep-th/04061900808.02271110.08031206.2630 | DOI | MR | DOI | MR | DOI | DOI | MR

[11] 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

[12] 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: ; “Dynamical structure factor at small $q$ for the $XXZ$ spin-$1/2$ chain”, J. Stat. Mech., 2007:8 (2007), P08022, 64 pp., arXiv: cond-mat/06036810706.4327 | DOI | DOI | MR

[13] 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

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

[15] N. A. Slavnov, Algebraic Bethe Ansatz and Correlation Functions: An Advanced Course, World Sci., Singapore, 2022 | DOI | MR

[16] W. Heisenberg, “Zur Theorie des Ferromagnetismus”, Z. Phys., 49:9–10 (1928), 619–636 | DOI

[17] B. Sutherland, “Two-dimensional hydrogen bonded crystals without the ice rule”, J. Math. Phys., 11:11 (1970), 3183–3186 | DOI

[18] C. Fan, F. Y. Wu, “General lattice model of phase transitions”, Phys. Rev. B, 2:3 (1970), 723–733 | DOI

[19] R. J. Baxter, “Eight-vertex model in lattice statistics”, Phys. Rev. Lett., 26:14 (1971), 832–833 | DOI

[20] R. Bekster, Tochno reshaemye modeli v statisticheskoi mekhanike, Mir, M., 1985 | MR | MR | Zbl

[21] S. Belliard, N. A. Slavnov, “Why scalar products in the algebraic Bethe ansatz have determinant representation”, JHEP, 10 (2019), 103, 16 pp., arXiv: 1908.00032 | DOI | MR

[22] N. Slavnov, A. Zabrodin, A. Zotov, “Scalar products of Bethe vectors in the 8-vertex model”, JHEP, 06 (2020), 123, 53 pp., arXiv: 2005.11224 | DOI | MR

[23] N. Kitanine, J.-M. Maillet, V. Terras, “Form factors of the XXZ Heisenberg spin-$1/2$ finite chain”, Nucl. Phys. B, 554:3 (1999), 647–678, arXiv: math-ph/9807020 | DOI | MR

[24] F. Göhmann, V. E. Korepin, “Solution of the quantum inverse problem”, J. Phys. A: Math. Gen., 33:6 (2000), 1199–1220, arXiv: hep-th/9910253 | DOI | MR

[25] J. M. Maillet, V. Terras, “On the quantum inverse scattering problem”, Nucl. Phys. B, 575:3 (2000), 627–644, arXiv: hep-th/9911030 | DOI | MR

[26] G. Kulkarni, N. A. Slavnov, “Deistviya elementov matritsy monodromii v obobschennom algebraicheskom anzatse Bete”, prinyato k publikatsii, TMF, arXiv: 2303.02439

[27] E. Lieb, T. Schultz, D. Mattis, “Two soluble models of an antiferromagnetic chain”, Ann. Phys., 16:3 (1961), 407–466 | DOI | MR

[28] B. M. McCoy, “Spin correlation functions of the $X$–$Y$ model”, Phys. Rev., 173:2 (1968), 531–541 | DOI

[29] Th. Niemeijer, “Some exact calculations on a chain of spins 1/2”, Physica, 36:3 (1967), 377–419 | DOI

[30] S. Katsura, T. Horiguchi, M. Suzuki, “Dynamical properties of the isotropic $XY$ model”, Physica, 46:1 (1970), 67–86 | DOI

[31] J. H. H. Perk, H. W. Capel, “Time-dependent $xx$-correlation functions in the one dimensional $XY$-model”, Phys. A, 89:2 (1977), 265–303 | DOI

[32] H. G. Vaidya, C. A. Tracy, “Crossover scaling function for the one-dimensional $XY$ model at zero temperature”, Phys. Lett. A, 68:3–4 (1978), 378–380 | DOI | MR

[33] T. Tonegawa, “Transverse spin correlation function of the one-dimensional spin-$1/2$ $XY$ model”, Solid State Comm., 40:11 (1981), 983–986 | DOI

[34] M. D'lorio, U. Glaus, E. Stoll, “Transverse spin dynamics of a one-dimensional $XY$ system: A fit to spin-spin relaxation data”, Solid State Commun., 47:5 (1983), 313–315 | DOI

[35] A. G. Izergin, N. A. Kitanin, N. A. Slavnov, “O korrelyatsionnykh funktsiyakh $XY$-modeli”, Voprosy kvantovoi teorii polya i statisticheskoi fiziki. 13, Zap. nauchn. sem. POMI, 224, POMI, SPb., 1995, 178–191 | DOI | MR | Zbl

[36] K. Fabricius, B. M. McCoy, “New developments in the eight vertex model”, J. Stat. Phys., 111:1–2 (2003), 323–337, arXiv: ; “New developments in the eight vertex model II. Chains of odd length”, 120:1–2 (2005), 37–70, arXiv: ; “Functional equations and fusion matrices for the eight-vertex model”, Publ. Res. Inst. Math. Sci., 40:3 (2004), 905–932, arXiv: cond-mat/0207177cond-mat/0410113cond-mat/0311122 | DOI | MR | DOI | MR | DOI | MR

[37] K. Fabricius, B. M. McCoy, “An elliptic current operator for the 8 vertex model”, J. Phys. A: Math. Gen., 39:48 (2006), 14869–14886, arXiv: cond-mat/0606190 | DOI | MR

[38] T. Deguchi, “The 8V CSOS model and the $sl_2$ loop algebra symmetry of the six-vertex model at roots of unity”, Internat. J. Modern Phys. B, 16:14–15 (2002), 1899–1905, arXiv: cond-mat/0110121 | DOI | MR

[39] K. Fabricius, “A new $Q$-matrix in the eight vertex model”, J. Phys. A: Math. Theor., 40:15 (2007), 4075–4086, arXiv: cond-mat/0610481 | DOI | MR

[40] S. Kharchev, A. Zabrodin, “Theta vocabulary I”, J. Geom. Phys., 94 (2015), 19–31, arXiv: 1502.04603 | DOI | MR