Publication: A new perspective on the integrability of Inozemtsev's elliptic spin chain
Loading...
Official URL
Full text at PDC
Publication Date
2014-12
Advisors (or tutors)
Editors
Journal Title
Journal ISSN
Volume Title
Publisher
Academic Press Inc Elsevier Science
Citations
Exportar
Abstract
The aim of this paper is studying from an alternative point of view the integrability of the spin chain with long-range elliptic interactions introduced by Inozemtsev. Our analysis relies on some well-established conjectures characterizing the chaotic vs. integrable behavior of a quantum system, formulated in terms of statistical properties of its spectrum. More precisely, we study the distribution of consecutive levels of the (unfolded) spectrum, the power spectrum of the spectral fluctuations, the average degeneracy, and the equivalence to a classical vertex model. Our results are consistent with the general consensus that this model is integrable, and that it is closer in this respect to the Heisenberg chain than to its trigonometric limit (the Haldane-Shastry chain). On the other hand, we present some numerical and analytical evidence showing that the level density of Inozemtsev's chain is asymptotically Gaussian as the number of spins tends to infinity, as is the case with the Haldane-Shastry chain. We are also able to compute analytically the mean and the standard deviation of the spectrum, showing that their asymptotic behavior coincides with that of the Haldane-Shastry chain.
Description
© 2014 Elsevier Inc. All rights reserved.
This work was supported in part by Spain's MINECO under grant no. FIS2011-22566.
UCM subjects
Unesco subjects
Keywords
Citation
[1] F. D. M. Haldane, Phys. Rev. Lett. 60 (1988) 635–638.
[2] B. S. Shastry, Phys. Rev. Lett. 60 (1988) 639–642.
[3] M. Fowler, J. A. Minahan, Phys. Rev. Lett. 70 (1993) 2325–2328.
[4] F. D. M. Haldane, Z. N. C. Ha, J. C. Talstra, D. Bernard, V. Pasquier, Phys. Rev. Lett. 69 (1992) 2021–2025.
[5] F. Finkel, A. González-López, Phys. Rev. B 72 (2005) 174411(6).
[6] B. Basu-Mallick, N. Bondyopadhaya, K. Hikami, SIGMA 6 (2010) 091(13).
[7] Z. N. C. Ha, F. D. M. Haldane, Phys. Rev. B 46 (1992) 9359–9368.
[8] A. P. Polychronakos, Phys. Rev. Lett. 70 (1993) 2329–2331.
[9] B. Sutherland, Phys. Rev. A 4 (1971) 2019–2021.
[10] B. Sutherland, Phys. Rev. A 5 (1972) 1372–1376.
[11] F. Calogero, J. Math. Phys. 12 (1971) 419–436.
[12] J. A. Minahan, A. P. Polychronakos, Phys. Lett. B 302 (1993) 265–270.
[13] V. I. Inozemtsev, D. V. Meshcheryakov, Phys. Scr. 33 (1986) 99–104.
[14] V. I. Inozemtsev, Phys. Scr. 53 (1996) 516–520.
[15] H. Frahm, J. Phys. A: Math. Gen. 26 (1993) L473–L479
[16] H. Frahm, V. I. Inozemtsev, J. Phys. A: Math. Gen. 27 (1994) L801–L807.
[17] M. A. Olshanetsky, A. M. Perelomov, Phys. Rep. 94 (1983) 313–404.
[18] J. Dittrich, V. I. Inozemtsev, J. Phys. A: Math. Gen. 26 (1993) L753–L756.
[19] G. Felder, A. Varchenko, Comp. Math. 107 (1997) 143–175.
[20] K. Takemura, Lett. Math. Phys. 53 (2000) 181–194.
[21] Y. Komori, K. Takemura, Commun. Math. Phys. 227 (2002) 93–118.
[22] E. Langmann, Lett. Math. Phys. 94 (2010) 63–75.
[23] J. Dittrich, V. I. Inozemtsev, Regul. Chaotic Dyn. 14 (2009) 218–222.
[24] J. C. Barba, V. I. Inozemtsev, Phys. Lett. A 372 (2008) 5951.
[25] V. I. Inozemtsev, J. Stat. Phys. 59 (1990) 1143–1155.
[26] D. Serban, M. Staudacher, JHEP 06 (2004) 001(31).
[27] T. Bargheer, N. Beisert, F. Loebbert, J. Phys. A: Math. Theor. 42 (2009) 285205(58).
[28] A. Rej, Lett. Math. Phys. 99 (2012) 85–102.
[29] D. Serban, JHEP 01 (2013) 012(16).
[30] V. I. Inozemtsev, Lett. Math. Phys. 36 (1996) 55–63.
[31] V. I. Inozemtsev, Phys. Part. Nucl. 34 (2003) 166–193.
[32] M. V. Berry, M. Tabor, Proc. R. Soc. London Ser. A 356 (1977) 375–394.
[33] O. Bohigas, M. J. Giannoni, C. Schmit, Phys. Rev. Lett. 52 (1984) 1–4.
[34] D. Poilblanc, T. Ziman, J. Bellissard, F. Mila, J. Montambaux, Europhys. Lett. 22 (1993) 537– 542.
[35] A. Relano, J. M. G. Gómez, R. A. Molina, J. Retamosa, E. Faleiro, Phys. Rev. Lett. 89 (2002) 244102(4).
[36] E. Faleiro, J. M. G. Gómez, R. A. Molina, L. Muñoz, A. Relaño, J. Retamosa, Phys. Rev. Lett. 93 (2004) 244101(4).
[37] D. Bernard, Int. J. Mod. Phys. B 7 (1993) 3517–3530.
[38] A. Enciso, F. Finkel, A. Gonzalez-López, Phys. Rev. E 82 (2010) 051117(6).
[39] B. Berndt, B. P. Yeap, Adv. Appl. Math. 29 (2002) 358–385.
[40] A. P. Polychronakos, Nucl. Phys. B 419 (1994) 553–566.
[41] F. W. J. Olver, D. W. Lozier, R. F. Boisvert, C. W. Clark (Eds.), NIST Handbook of Mathematical Functions, Cambridge Univeristy Press, 2010.
[42] A. P. Polychronakos, J. Phys. A: Math. Gen. 39 (2006) 12793–12845.
[43] A. Enciso, F. Finkel, A. Gonzalez-López, M. A. Rodríguez, Nucl. Phys. B 707 (2005) 553–576.
[44] B. Basu-Mallick, N. Bondyopadhaya, Nucl. Phys. B 757 (2006) 280–302.
[45] J. C. Barba, F. Finkel, A. Gonzalez-López, M. A. Rodríguez, Phys. Rev. B 77 (2008) 214422(10).
[46] J. C. Barba, F. Finkel, A. González-López, M. A. Rodríguez, Nucl. Phys. B 806 (2009) 684– 714.
[47] B. Basu-Mallick, F. Finkel, A. Gonzalez-López, Nucl. Phys. B 812 (2009) 402–423.
[48] B. Basu-Mallick, N. Bondyopadhaya, Phys. Lett. A 373 (2009) 2831–2836.
[49] B. Basu-Mallick, F. Finkel, A. Gonzalez-López, Nucl. Phys. B 866 (2013) 391–413.
[50] M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, Dover, New York, ninth edition, 1970.
[51] E. M. Stein, R. Shakarchi, Complex Analysis, Princeton University Press, Princeton, N.J., 2003.
[52] F. Haake, Quantum Signatures of Chaos, Springer-Verlag, Berlin, second edition, 2001.
[53] M. L. Mehta, Random Matrices, Elsevier, San Diego, 3rd edition, 2004.
[54] J.-C. A. d’Auriac, J.-M. Maillard, C. M. Viallet, J. Phys. A: Math. Gen. 35 (2002) 4801–4822.
[55] J. C. Barba, F. Finkel, A. Gonzalez-López, M. A. Rodríguez, Europhys. Lett. 83 (2008) 27005(6).
[56] M. S. Santhanam, J. N. Bandyopadhyay, Phys. Rev. Lett. 95 (2005) 114101(4).
[57] C. Male, G. Le Caër, R. Delannay, Phys. Rev. E 76 (2007) 042101(4).
[58] A. Relano, Phys. Rev. Lett. 100 (2008) 224101(4).
[59] J. C. Barba, F. Finkel, A. González-López, M. A. Rodríguez, Nucl. Phys. B 839 (2010) 499–525.
[60] K. Hikami, Nucl. Phys. B 441 (1995) 530–548.
[61] F. Finkel, A. González-López, 2014. To be published.
[62] E. P. Miles, Jr, Am. Math. Mon. 67 (1960) 745–752.
[63] D. F. Lawden, Elliptic Functions and Applications, Springer-Verlag, Berlin, 1989.