Abstract
In this note, we propose an expression for the eigenvectors and scalar products for a class of spin chains with long-range interaction and su(2) symmetry. This class includes the Inozemtsev spin chain as well as the BDS spin chain, which is a reduction of the one-dimensional Hubbard model at half-filling to the spin sector. The proposal is valid for large spin chains and is based on the construction of the monodromy matrix using the Dunkl operators. For the Inozemtsev model these operators are known explicitly. This construction gives in particular the eigenvectors of (an operator closely related to) the dilatation operator of the \( \mathcal{N}=4 \) gauge theory in the su(2) sector up to three-loop order, as well as their scalar products. We suggest how this will affect the expression for the quasi classical limit of the three-point functions obtained by I. Kostov and how to include the all-loop interaction.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [Int. J. Theor. Phys. 38 (1999) 1113] [hep-th/9711200] [INSPIRE].
A.M. Polyakov, The wall of the cave, Int. J. Mod. Phys. A 14 (1999) 645 [hep-th/9809057] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
D. Serban, Integrability and the AdS/CFT correspondence, J. Phys. A 44 (2011) 124001 [arXiv:1003.4214] [INSPIRE].
N. Beisert et al., Review of AdS/CFT integrability: an overview, Lett. Math. Phys. 99 (2012) 3 [arXiv:1012.3982] [INSPIRE].
R. Roiban and A. Volovich, Yang-Mills correlation functions from integrable spin chains, JHEP 09 (2004) 032 [hep-th/0407140] [INSPIRE].
J. Escobedo, N. Gromov, A. Sever and P. Vieira, Tailoring three-point functions and integrability, JHEP 09 (2011) 028 [arXiv:1012.2475] [INSPIRE].
J. Escobedo, N. Gromov, A. Sever and P. Vieira, Tailoring three-point functions and integrability II. Weak/strong coupling match, JHEP 09 (2011) 029 [arXiv:1104.5501] [INSPIRE].
N. Gromov, A. Sever and P. Vieira, Tailoring three-point functions and integrability III. Classical tunneling, JHEP 07 (2012) 044 [arXiv:1111.2349] [INSPIRE].
A. Bissi, T. Harmark and M. Orselli, Holographic 3-point function at one loop, JHEP 02 (2012) 133 [arXiv:1112.5075] [INSPIRE].
N. Gromov and P. Vieira, Quantum integrability for three-point functions, arXiv:1202.4103 [INSPIRE].
L.F. Alday, D. Gaiotto and J. Maldacena, Thermodynamic bubble ansatz, JHEP 09 (2011) 032 [arXiv:0911.4708] [INSPIRE].
L.F. Alday, J. Maldacena, A. Sever and P. Vieira, Y-system for Scattering Amplitudes, J. Phys. A 43 (2010) 485401 [arXiv:1002.2459] [INSPIRE].
N. Drukker, Integrable Wilson loops, arXiv:1203.1617 [INSPIRE].
D. Correa, J. Maldacena and A. Sever, The quark anti-quark potential and the cusp anomalous dimension from a TBA equation, JHEP 08 (2012) 134 [arXiv:1203.1913][INSPIRE].
R.A. Janik and P. Laskos-Grabowski, Surprises in the AdS algebraic curve constructions: Wilson loops and correlation functions, Nucl. Phys. B 861 (2012) 361 [arXiv:1203.4246] [INSPIRE].
K. Okuyama and L.-S. Tseng, Three-point functions in N = 4 SYM theory at one-loop, JHEP 08 (2004) 055 [hep-th/0404190] [INSPIRE].
A.G. Izergin, Partition function of the six-vertex model in a finite volume, Sov. Phys. Dokl. 32 (1987)878.
V.E. Korepin, Calculation of norms of Bethe wave functions, Commun. Math. Phys. 86 (1982) 391 [INSPIRE].
N.A. Slavnov, The algebraic Bethe ansatz and quantum integrable systems, Russ. Math. Surv. 62 (2007) 727.
O. Foda, N = 4 SYM structure constants as determinants, JHEP 03 (2012) 096 [arXiv:1111.4663] [INSPIRE].
I. Kostov, Classical Limit of the Three-Point Function from Integrability, to appear.
N. Beisert, C. Kristjansen and M. Staudacher, The dilatation operator of conformal N = 4 super Yang-Mills theory, Nucl. Phys. B 664 (2003) 131 [hep-th/0303060] [INSPIRE].
D. Serban and M. Staudacher, Planar \( \mathcal{N}=4 \) gauge theory and the Inozemtsev long range spin chain, JHEP 06 (2004) 001 [hep-th/0401057] [INSPIRE].
V. Inozemtsev, Integrable Heisenberg-van Vleck chains with variable range exchange, Phys. Part. Nucl. 34 (2003) 166 [hep-th/0201001] [INSPIRE].
N. Beisert, V. Dippel and M. Staudacher, A novel long range spin chain and planar N = 4 super Yang-Mills, JHEP 07 (2004) 075 [hep-th/0405001] [INSPIRE].
C.F. Dunkl, Differential-Difference Operators Associated to Reflection Groups, Trans. Am. Math. Soc. 311 (1989) 167.
A.P. Polychronakos, Exchange operator formalism for integrable systems of particles, Phys. Rev. Lett. 69 (1992) 703 [hep-th/9202057] [INSPIRE].
V. Braun, Y.-H. He, B.A. Ovrut and T. Pantev, The exact MSSM spectrum from string theory, JHEP 05 (2006) 043 [hep-th/0512177] [INSPIRE].
T. Bargheer, N Beisert and F. Loebbert, Boosting Nearest-Neighbour to Long-Range Integrable Spin Chains, J. Stat. Mech. 0811 (2008) L11001 [arXiv:0807.5081].
A.P. Polychronakos, Lattice integrable systems of Haldane-Shastry type, Phys. Rev. Lett. 70 (1993) 2329 [hep-th/9210109] [INSPIRE].
D. Bernard, M. Gaudin, F. Haldane and V. Pasquier, Yang-Baxter equation in spin chains with long range interactions, hep-th/9301084 [INSPIRE].
J.C. Talstra and F.D.M. Haldane, Integrals of motion of the Haldane-Shastry model, J. Phys. A 28 (1995) 2369 [cond-mat/9411065].
N. Gromov, Quantum Integrability for Three-Point Functions, talk at the Perimeter Institute, Waterloo Canada (2012), http://pirsa.org/displayFlash.php?id=12020162.
D. Bernard, V. Pasquier and D. Serban, Exact Solution of Long-Range Interacting Spin Chains with Boundaries, Europhys. Lett. 30 (1995) 301.
D. Serban, in progress.
L. Faddeev, How algebraic Bethe ansatz works for integrable model, hep-th/9605187 [INSPIRE].
M. Wheeler, An Izergin-Korepin procedure for calculating scalar products in six-vertex models, Nucl. Phys. B 852 (2011) 468 [arXiv:1104.2113] [INSPIRE].
F. Göhmann and V. Korepin, The Hubbard chain: Lieb-Wu equations and norm of the eigenfunctions, Phys. Lett. A 263 (1999) 293 [cond-mat/9908114] [INSPIRE].
N. Beisert, B. Eden and M. Staudacher, Transcendentality and crossing, J. Stat. Mech. 0701 (2007) P01021 [hep-th/0610251] [INSPIRE].
V. Kazakov, A. Marshakov, J. Minahan and K. Zarembo, Classical/quantum integrability in AdS/CFT, JHEP 05 (2004) 024 [hep-th/0402207] [INSPIRE].
N. Beisert, V. Kazakov, K. Sakai and K. Zarembo, The algebraic curve of classical superstrings on AdS 5 × S 5, Commun. Math. Phys. 263 (2006) 659 [hep-th/0502226] [INSPIRE].
N. Gromov, Y-system and quasi-classical strings, JHEP 01 (2010) 112 [arXiv:0910.3608] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1203.5842
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Serban, D. A note on the eigenvectors of long-range spin chains and their scalar products. J. High Energ. Phys. 2013, 12 (2013). https://doi.org/10.1007/JHEP01(2013)012
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP01(2013)012