Research PapersNo Access

ANALYTICAL BETHE ANSATZ FOR OPEN SPIN CHAINS WITH SOLITON NONPRESERVING BOUNDARY CONDITIONS

Laboratoire d'Annecy-le-Vieux de Physique Théorique, LAPTH, CNRS, UMR 5108, Université de Savoie, B.P. 110, F-74941 Annecy-le-Vieux Cedex, France

Search for more papers by this author

A. DOIKOU

Laboratoire d'Annecy-le-Vieux de Physique Théorique, LAPTH, CNRS, UMR 5108, Université de Savoie, B.P. 110, F-74941 Annecy-le-Vieux Cedex, France

Search for more papers by this author

L. FRAPPAT

Laboratoire d'Annecy-le-Vieux de Physique Théorique, LAPTH, CNRS, UMR 5108, Université de Savoie, B.P. 110, F-74941 Annecy-le-Vieux Cedex, France

Institut Universitaire de France, 103, Bd Saint-Michel 75005 Paris, France

Search for more papers by this author

É. RAGOUCY

Laboratoire d'Annecy-le-Vieux de Physique Théorique, LAPTH, CNRS, UMR 5108, Université de Savoie, B.P. 110, F-74941 Annecy-le-Vieux Cedex, France

Search for more papers by this author

, and

N. CRAMPÉ

University of York, Department of Mathematics, Heslington, York YO10 5DD, UK

Search for more papers by this author

https://doi.org/10.1142/S0217751X06029077Cited by:12 (Source: Crossref)

Abstract

We present an "algebraic treatment" of the analytical Bethe ansatz for open spin chains with soliton nonpreserving (SNP) boundary conditions. For this purpose, we introduce abstract monodromy and transfer matrices which provide an algebraic framework for the analytical Bethe ansatz. It allows us to deal with a generic open SNP spin chain possessing on each site an arbitrary representation. As a result, we obtain the Bethe equations in their full generality. The classification of finite dimensional irreducible representations for the twisted Yangians are directly linked to the calculation of the transfer matrix eigenvalues.

In memory of our friend Daniel Arnaudon

Keywords:

PACS: 02.20.Uw, 03.65.Fd, 75.10.Pq

You currently do not have access to the full text article.
Recommend the journal to your library today!

References

A. Gleizes and M. Verdaguer, J. Am. Chem. Soc. 103, 7373 (1981), DOI: 10.1021/ja00414a074. Web of Science, Google Scholar
D. C. Johnstonet al., Phys. Rev. B 35, 219 (1987), DOI: 10.1103/PhysRevB.35.219. Web of Science, ADS, Google Scholar
T. Barnes and J. Riera, Phys. Rev. B 50, 6817 (1994), DOI: 10.1103/PhysRevB.50.6817. Web of Science, ADS, Google Scholar
A. V. Prokofiev, F. Büllesfeld, W. Assmus, H. Schwenk, D. Wichert, U. Löw and B. Lüthi, Magnetic properties of the low dimensional spin system (VO)2P2O7: ESR and susceptibility, cond-mat/9712137 . Google Scholar
A. Weiße, G. Bouzerar and H. Fehske, Eur. Phys. J. B 7, 5 (1999). Web of Science, ADS, Google Scholar
J. Kikuchiet al., Phys. Rev. B 60, 6731 (1999), DOI: 10.1103/PhysRevB.60.6731. Web of Science, ADS, Google Scholar
E. Janod, L. Leonyuk and V. Maltsev, Experimental evidence for a spin gap in the s = 1/2 quantum antiferromagnet Cu2(OH)2CO3, cond-mat/0009040 . Google Scholar
M. C. Gutzwiller, Phys. Rev. Lett. 10, 159 (1963), DOI: 10.1103/PhysRevLett.10.159. Web of Science, ADS, Google Scholar
B. M. McCoy and T. T. Wu, Phys. Lett. B 87, 50 (1979), DOI: 10.1016/0370-2693(79)90015-7. Web of Science, ADS, Google Scholar
I. Affleck, Fields, Strings, and Critical Phenomena, Les Houches Lectures, 1988, eds. E. Brezin and J. Zinn-Justin (Elsevier, 1990) p. 563. Google Scholar
A. B. Zamolodchikov and A. B. Zamolodchikov, Nucl. Phys. B 379, 602 (1992), DOI: 10.1016/0550-3213(92)90136-Y. Web of Science, ADS, Google Scholar
P. Fendley, H. Saleur and A. B. Zamolodchikov, Int. J. Mod. Phys. A 8, 5751 (1993), DOI: 10.1142/S0217751X93002277. Link, Web of Science, ADS, Google Scholar
L. Lipatov, JETP Lett. 59, 596 (1994). Web of Science, ADS, Google Scholar
L. Faddeev and G. Korchemsky, Phys. Lett. B 342, 311 (1995), DOI: 10.1016/0370-2693(94)01363-H. Web of Science, ADS, Google Scholar
J. A. Minahan and K. Zarembo, J. High Energy Phys. 0303, 013 (2003). ADS, Google Scholar
N. Beisert and M. Staudacher, Nucl. Phys. B 670, 439 (2003), DOI: 10.1016/j.nuclphysb.2003.08.015. Web of Science, ADS, Google Scholar
G. Ferretti, R. Heise and K. Zarembo, New integrable structures in large-N QCD, hep-th/0404187 . Google Scholar
I. V. Cherednik, Theor. Math. Phys. 61, 977 (1984), DOI: 10.1007/BF01038545. Web of Science, Google Scholar
E. K. Sklyanin, J. Phys. A 21, 2375 (1988), DOI: 10.1088/0305-4470/21/10/015. ADS, Google Scholar
A. Doikou, J. Phys. A 33, 8797 (2000), DOI: 10.1088/0305-4470/33/48/315. ADS, Google Scholar
D. Arnaudonet al., J. Stat. Mech.: Theor. Exp. 08, P005 (2004). Google Scholar
A. Doikou, On reflection algebras and twisted Yangians, hep-th/0403277 . Google Scholar
G. I. Olshanski, Sov. Math. Dokl. 36, 569 (1988). Web of Science, Google Scholar
G. I. Olshanski, Representations of infinite dimensional classical groups, limits enveloping algebras and Yangians, in Topics in Representation Theory; A. A. Kirillov (ed.), Adv. Sov. Math. 2, 1 (1998) . Google Scholar
A. Molev, M. Nazarov and G. Olshanski, Russian Math. Survey 51, 205 (1996), DOI: 10.1070/RM1996v051n02ABEH002772. Web of Science, ADS, Google Scholar
P. Bowcocket al., Nucl. Phys. B 445, 469 (1995), DOI: 10.1016/0550-3213(95)00153-J. Web of Science, ADS, Google Scholar
P. Bowcock, E. Corrigan and R. H. Rietdijk, Nucl. Phys. B 465, 350 (1996), DOI: 10.1016/0550-3213(96)00050-8. Web of Science, ADS, Google Scholar
G. M. Gandenberger, New nondiagonal solutions to the $A^{(1)}_n$ boundary Yang–Baxter equation, hep-th/9911178 . Google Scholar
G. W. Delius, J. High Energy Phys. 9809, 016 (1998). ADS, Google Scholar
G. W. Delius and N. J. MacKay, Commun. Math. Phys. 233, 173 (2003), DOI: 10.1007/s00220-002-0758-4. Web of Science, ADS, Google Scholar
D. Arnaudon, N. Crampé, A. Doikou, L. Frappat and E. Ragoucy, Analytical Bethe ansatz for closed and open gl(n)-spin chains in any representation, math-ph/0411021 . Google Scholar
A. I. Molev, J. Math. Phys. 39, 5559 (1998), DOI: 10.1063/1.532551. Web of Science, ADS, Google Scholar
C. N. Yang, Rev. Lett. 19, 1312 (1967), DOI: 10.1103/PhysRevLett.19.1312. Web of Science, ADS, Google Scholar
R. J. Baxter, Ann. Phys. 70, 193 (1972); ibid., J. Stat. Phys. 8, 25 (1973); ibid., Exactly Solved Models in Statistical Mechanics (Academic Press, 1982) . Google Scholar
J. B. McGuire, J. Math. Phys. 5, 622 (1964), DOI: 10.1063/1.1704156. Web of Science, ADS, Google Scholar
V. E. Korepin, Commun. Math. Phys. 76, 165 (1980), DOI: 10.1007/BF01212824. Web of Science, ADS, Google Scholar
V. E. Korepin , G. Izergin and N. M. Bogoliubov , Quantum Inverse Scattering Method, Correlation Functions and Algebraic Bethe Ansatz ( Cambridge University Press , 1993 ) . Google Scholar
V. G. Drinfel'd, Sov. Math. Dokl. 32, 254 (1985). Web of Science, Google Scholar
V. G. Drinfel'd, Sov. Math. Dokl. 36, 212 (1988). Web of Science, Google Scholar
L. D. Faddeev, N. Yu. Reshetikhin and L. A. Takhtajan, Leningrad Math. J. 1, 193 (1990). Web of Science, Google Scholar
A. I. Molev, Handbook of Algebra 3, ed. M. Hazewinkel (Elsevier, 2003) p. 907, math-QA/0211288. Google Scholar
M. Nazarov and V. Tarasov, Int. Math. Res. Not. 3, 125 (1998), DOI: 10.1155/S1073792898000129. Google Scholar
A. I. Molev, Duke Math. J. 112, 307 (2002), DOI: 10.1215/S0012-9074-02-11224-1. Web of Science, Google Scholar