Geodesic
Action of the monodromy matrix elements in the generalized algebraic Bethe ansatz
Teoretičeskaâ i matematičeskaâ fizika, Tome 217 (2023) no. 3, pp. 555-576 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 calculate the actions of monodromy matrix elements on Bethe vectors as linear combinations of new Bethe vectors. We also compute the multiple action of the gauge-transformed monodromy matrix elements on the pre-Bethe vector and express the results in terms of the partition function of the $8$-vertex model.
Keywords: generalized algebraic Bethe ansatz, Bethe vectors, gauge transformed monodromy matrix, domain-wall partition function. 
@article{TMF_2023_217_3_a8,
     author = {G. Kulkarni and N. A. Slavnov},
     title = {Action of the~monodromy matrix elements in the~generalized algebraic {Bethe} ansatz},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {555--576},
     year = {2023},
     volume = {217},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2023_217_3_a8/}
}
TY  - JOUR
AU  - G. Kulkarni
AU  - N. A. Slavnov
TI  - Action of the monodromy matrix elements in the generalized algebraic Bethe ansatz
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2023
SP  - 555
EP  - 576
VL  - 217
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/TMF_2023_217_3_a8/
LA  - ru
ID  - TMF_2023_217_3_a8
ER  - 
%0 Journal Article
%A G. Kulkarni
%A N. A. Slavnov
%T Action of the monodromy matrix elements in the generalized algebraic Bethe ansatz
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2023
%P 555-576
%V 217
%N 3
%U http://geodesic.mathdoc.fr/item/TMF_2023_217_3_a8/
%G ru
%F TMF_2023_217_3_a8
G. Kulkarni; N. A. Slavnov. Action of the monodromy matrix elements in the generalized algebraic Bethe ansatz. Teoretičeskaâ i matematičeskaâ fizika, Tome 217 (2023) no. 3, pp. 555-576. http://geodesic.mathdoc.fr/item/TMF_2023_217_3_a8/

[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: hep-th/0406190 | DOI | MR

[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: ; N. Kitanine, K. K. Kozlowski, J. M. Maillet, N. A. Slavnov, V. Terras, “A form factor approach to the asymptotic behavior of correlation functions”, 2011, no. 12, 2011, P12010, 28 pp., arXiv: ; “Form factor approach to dynamical correlation functions in critical models”, 2012, no. 9, 2012, P09001, 33 pp., arXiv: 0808.02271110.08031206.2630 | DOI | MR | DOI | DOI | MR

[12] 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

[13] 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

[14] 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

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

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

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

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

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

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

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

[22] 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

[23] 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

[24] 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

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

[26] 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

[27] 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

[28] 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

[29] S. Belliard, S. Pakuliak, E. Ragoucy, N. A. Slavnov, “Bethe vectors of quantum integrable models with $GL(3)$ trigonometric $R$-matrix”, SIGMA, 9 (2013), 058, 23 pp., arXiv: 1304.7602 | MR

[30] S. Pakuliak, V. Rubtsov, A. Silantyev, “The SOS model partition function and the elliptic weight functions”, J. Phys. A: Math. Theor., 41:29 (2008), 295204, 20 pp., arXiv: 0802.0195 | DOI | MR

[31] W-L. Yang, Y.-Z. Zhang, “Partition function of the eight-vertex model with domain wall boundary condition”, J. Math. Phys., 50:8 (2009), 083518, 14 pp., arXiv: 0903.3089 | DOI | MR

[32] H. Rosengren, “An Izergin–Korepin-type identity for the 8VSOS model, with applications to alternating sign matrices”, Adv. Appl. Math., 43:2 (2009), 137–155, arXiv: 0801.1229 | DOI | MR

[33] G. Felder, “Elliptic quantum groups”, Proceedings of XIth International Congress of Mathematical Physics (July 18 – 22, 1994, Paris, France), International Press, Cambridge, MA, 1995, 211–218, arXiv: hep-th/9412207 | MR

[34] A. Hutsalyuk, A. Lyashik, S. Z. Pakuliak, E. Ragoucy, N. A. Slavnov, “Multiple actions of the monodromy matrix in $\mathfrak{gl}(2|1)$-invariant integrable models”, SIGMA, 12 (2016), 099, 22 pp., arXiv: 1605.06419 | MR

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

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

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