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Uspekhi Mat. Nauk, 2007, Volume 62, Issue 4(376), Pages 91–132 (Mi umn6847)  

This article is cited in 43 scientific papers (total in 43 papers)

The algebraic Bethe ansatz and quantum integrable systems

N. A. Slavnov

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: Methods are considered for applying an algebra with bilinear commutation relations to the theory of quantum integrable systems. This survey describes most of the results obtained in this area over the last twenty years, mainly in connection with the computation of correlation functions of quantum integrable systems. Methods for constructing eigenfunctions of the quantum transfer matrix and computing inner products and correlation functions are presented in detail. An example of application of the general scheme to the model of the $XXZ$ Heisenberg chain is considered.

DOI: https://doi.org/10.4213/rm6847

Full text: PDF file (840 kB)
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English version:
Russian Mathematical Surveys, 2007, 62:4, 727–766

Bibliographic databases:

UDC: 517.958+530.145
MSC: Primary 82B23; Secondary 81R50, 81U40
Received: 22.03.2007

Citation: N. A. Slavnov, “The algebraic Bethe ansatz and quantum integrable systems”, Uspekhi Mat. Nauk, 62:4(376) (2007), 91–132; Russian Math. Surveys, 62:4 (2007), 727–766

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Karasev M.V., “Quantum geometry and quantum mechanics of integrable systems”, Russ. J. Math. Phys., 16:1 (2009), 81–92  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
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    5. Kostov I., “Classical limit of the three-point function of $N=4$ supersymmetric Yang-Mills theory from integrability”, Phys. Rev. Lett., 108:26 (2012), 261604, 5 pp.  crossref  adsnasa  isi  elib  scopus
    6. Murg V., Korepin V., Verstraete F., “Algebraic Bethe ansatz and tensor networks”, Phys. Rev. B, 86:4 (2012), 045125, 17 pp.  crossref  adsnasa  isi  elib  scopus
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    12. S Belliard, S Pakuliak, E Ragoucy, N.A. Slavnov, “Form factors inSU(3)-invariant integrable models”, J. Stat. Mech, 2013:04 (2013), P04033  crossref  mathscinet  isi  elib  scopus
    13. J.F. van Diejen, E. Emsiz, “Discrete harmonic analysis on a Weyl alcove”, Journal of Functional Analysis, 2013  crossref  mathscinet  isi  scopus
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    15. Omar Foda, Yunfeng Jiang, Ivan Kostov, Didina Serban, “A tree-level 3-point function in the su(3)-sector of planar $ \mathcal{N}=4 $ SYM”, J. High Energ. Phys, 2013:10 (2013)  crossref  isi  scopus
    16. N.M. Bogoliubov, C. Malyshev, “Correlation functions of XX0 Heisenberg chain, q-binomial determinants, and random walks”, Nuclear Physics B, 2013  crossref  mathscinet  isi  scopus
    17. A. Gorsky, A. Zabrodin, A. Zotov, “Spectrum of quantum transfer matrices via classical many-body systems”, J. High Energ. Phys, 2014:1 (2014)  crossref  mathscinet  isi  scopus
    18. Satoshi Okuda, Yutaka Yoshida, “G/G gauged WZW-matter model, Bethe Ansatz for q-boson model and Commutative Frobenius algebra”, J. High Energ. Phys, 2014:3 (2014)  crossref  mathscinet  isi  scopus
    19. A. Levin, M. Olshanetsky, A. Zotov, “Relativistic classical integrable tops and quantum R-matrices”, J. High Energ. Phys, 2014:7 (2014)  crossref  isi  scopus
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    21. S. Pakuliak, E. Ragoucy, N.A. Slavnov, “Zero modes method and form factors in quantum integrable models”, Nuclear Physics B, 2015  crossref  mathscinet  isi  scopus
    22. Rouven Frassek, “Algebraic Bethe ansatz forQ-operators: the Heisenberg spin chain”, J. Phys. A: Math. Theor, 48:29 (2015), 294002  crossref  mathscinet  zmath  isi  scopus
    23. Stanislav Pakuliak, Eric Ragoucy, Nikita A. Slavnov, “$GL(3)$-Based Quantum Integrable Composite Models. I. Bethe Vectors”, SIGMA, 11 (2015), 063, 20 pp.  mathnet  crossref  mathscinet  elib
    24. Stanislav Pakuliak, Eric Ragoucy, Nikita A. Slavnov, “$GL(3)$-Based Quantum Integrable Composite Models. II. Form Factors of Local Operators”, SIGMA, 11 (2015), 064, 18 pp.  mathnet  crossref  mathscinet  elib
    25. N. M. Bogolyubov, K. L. Malyshev, “Integrable models and combinatorics”, Russian Math. Surveys, 70:5 (2015), 789–856  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    26. de Leeuw M., Kristjansen Ch., Zarembo K., “One-Point Functions in Defect CFT and Integrability”, no. 8, 2015, 098  crossref  mathscinet  isi  scopus
    27. Burdik C., Fuksa J., Isaev A.P., Krivonos S.O., Navratil O., “Remarks Towards the Spectrum of the Heisenberg Spin Chain Type Models”, 46, no. 3, 2015, 277–309  crossref  mathscinet  isi  scopus
    28. Deguchi T., Giri P.R., “Exact quantum numbers of collapsed and non-collapsed two-string solutions in the spin-1/2 Heisenberg spin chain”, J. Phys. A-Math. Theor., 49:17 (2016), 174001  crossref  mathscinet  zmath  isi  elib  scopus
    29. Hutsalyuk A. Liashyk A. Pakuliak S.Z. Ragoucy E. Slavnov N.A., Nucl. Phys. B, 911 (2016), 902–927  crossref  mathscinet  zmath  isi  elib  scopus
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    31. Jan Fuksa, “Bethe Vectors for Composite Models with $\mathfrak{gl}(2|1)$ and $\mathfrak{gl}(1|2)$ Supersymmetry”, SIGMA, 13 (2017), 015, 17 pp.  mathnet  crossref
    32. N. A. Slavnov, “Algebraicheskii anzats Bete”, Lekts. kursy NOTs, 27, MIAN, M., 2017, 3–189  mathnet  crossref  mathscinet  elib
    33. Fuksa J., “On the Structure of Bethe Vectors”, Phys. Part. Nuclei Lett., 14:4 (2017), 624–630  crossref  isi  scopus
    34. J. Fuksa, N. A. Slavnov, “Form factors of local operators in supersymmetric quantum integrable models”, J. Stat. Mech. Theory Exp., 2017, 043106, 21 pp.  crossref  mathscinet  isi  scopus
    35. Mina Aganagic, Andrei Okounkov, “Quasimap counts and Bethe eigenfunctions”, Mosc. Math. J., 17:4 (2017), 565–600  mathnet  crossref
    36. Tarasov V., “Completeness of the Bethe Ansatz For the Periodic Isotropic Heisenberg Model”, Rev. Math. Phys., 30:8, SI (2018), 1840018  crossref  mathscinet  isi  scopus
    37. Gerrard A. MacKay N. Regelskis V., “Nested Algebraic Bethe Ansatz For Open Spin Chains With Even Twisted Yangian Symmetry”, Ann. Henri Poincare, 20:2 (2019), 339–392  crossref  mathscinet  zmath  isi  scopus
    38. Maulik D. Okounkov A., “Quantum Groups and Quantum Cohomology”, Asterisque, 2019, no. 408, 1+  crossref  isi
    39. Basso B., Coronado F., Komatsu Sh., Lam H.T., Vieira P., Zhong D.-l., “Asymptotic Four Point Functions”, J. High Energy Phys., 2019, no. 7, 082  crossref  isi
    40. Gerrard A., Regelskis V., “Nested Algebraic Bethe Ansatz For Orthogonal and Symplectic Open Spin Chains”, Nucl. Phys. B, 952 (2020), 114909  crossref  isi
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    43. Gerrard A., Regelskis V., “Nested Algebraic Bethe Ansatz For Deformed Orthogonal and Symplectic Spin Chains”, Nucl. Phys. B, 956 (2020), 115021  crossref  mathscinet  isi
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