Coordinate Bethe Ansatz for Spin $s$ XXX Model

Nicolas Crampé ; Eric Ragoucy ; Ludovic Alonzi

Symmetry, integrability and geometry: methods and applications, Tome 7 (2011)

Résumé

We compute the eigenfunctions and eigenvalues of the periodic integrable spin $s$ XXX model using the coordinate Bethe ansatz. To do so, we compute explicitly the Hamiltonian of the model. These results generalize what has been obtained for spin $\frac12$ and spin 1 chains.

Détail

Keywords: coordinate Bethe ansatz; spin chains.

Bibliographie

[1] Heisenberg W., “Zur Theorie des Ferromagnetismus”, Z. Phys., 49 (1928), 619–636 | DOI | Zbl

[2] Bethe H., “Zur Theorie der Metalle. I. Eigenwerte und Eigenfunktionen der linearen Atomkette”, Z. Phys., 71 (1931), 205–226 | DOI

[3] Kulish P. P., Sklyanin E. K., “Quantum inverse scattering method and the Heisenberg ferromagnet”, Phys. Lett. A, 70 (1979), 461–463 | DOI | MR

[4] Takhtajan L. A., Faddeev L. D., “The quantum method of the inverse problem and the Heisenberg XYZ model”, Russ. Math. Surveys, 34 (1979), 11–68 | DOI

[5] Sklyanin E. K., “Quantum inverse scattering method. Selected topics”, Quantum Group and Quantum Integrable Systems, Nankai Lectures Math. Phys., ed. Mo-Lin Ge, World Sci. Publ., River Edge, NJ, Singapore, 1992, 63–97, arXiv: hep-th/9211111 | MR

[6] Vichirko V. I., Reshetikhin N. Yu., “Excitation spectrum of the anisotropic generalization of an $\mathrm{SU}(3)$ magnet”, Theoret. and Math. Phys., 56 (1983), 805–812 ; Reshetikhin N. Yu., “A method of functional equations in the theory of exactly solvable quantum systems”, Lett. Math. Phys., 7 (1983), 205–213 ; Reshetikhin N. Yu., “The functional equation method in the theory of exactly soluble quantum systems”, Sov. Phys. JETP, 57 (1983), 691–696 ; Reshetikhin N. Yu., “Integrable models of quantum one-dimensional magnets with $O(n)$ and $Sp(2k)$ symmetry”, Theoret. and Math. Phys., 63 (1985), 555–569 ; Reshetikhin N. Yu., “The spectrum of the transfer matrices connected with Kac–Moody algebras”, Lett. Math. Phys., 14 (1987), 235–246 | DOI | MR | DOI | MR | MR | DOI | DOI | MR | Zbl

[7] Kulish P. P., Reshetikhin N. Y., Sklyanin E. K., “Yang–Baxter equation and representation theory. I”, Lett. Math. Phys., 5 (1981), 393–403 | DOI | MR | Zbl

[8] Faddeev L. D., “How algebraic Bethe ansatz works for integrable model”, Symétries Quantiques (Les Houches, 1995), Les Houches Summerschool Proceedings, 64, eds. A. Connes, K. Gawedzki and J. Zinn-Justin, North-Holland, Amsterdam, 1998, 149–219, arXiv: hep-th/9605187 | MR

[9] Lima-Santos A., “Bethe ansätze for 19-vertex models”, J. Phys. A: Math. Gen., 32 (1999), 1819–1839, arXiv: hep-th/9807219 | DOI | MR | Zbl

[10] Takhtajan L. A., “Introduction to algebraic Bethe ansatz”, Exactly Solvable Problems in Condensed Matter and Field Theory, Lecture Notes in Physics, 242, eds. B. S. Shastry, S. S. Jha and V. Singh, Springer, Berlin, Heidelberg, 1985, 175–220 | MR

[11] Zamolodchikov A. B., Fateev V. A., “A model factorized $S$-matrix and an integrable spin-1 Heisenberg ferromagnet”, Soviet J. Nuclear Phys., 32 (1980), 298–303 ; Takhtajan L. A., “The picture of low-lying excitations in the isotropic Heisenberg chain of arbitrary spins”, Phys. Lett. A, 87 (1982), 479–482 ; Babujian H., “Exact solution of the isotropic Heisenberg chain with arbitrary spins: thermodynamics of the model”, Nuclear Phys. B, 215 (1983), 317–336 | MR | DOI | MR | DOI | MR

[12] Gaudin M., La fonction d'onde de Bethe, Masson, Paris, 1983 | MR | Zbl

[13] Essler F. H. L., Frahm H., Göhmann F., Klümper A., Korepin V. E., The one-dimensional Hubbard model, Cambridge University Press, Cambridge, 2005 | Zbl

[14] Izergin A. G., Korepin V. E., “The quantum inverse scattering approach to correlation functions”, Comm. Math. Phys., 94 (1984), 67–97 | DOI | MR

[15] Izergin A. G., Korepin V. E., Reshetikhin N. Yu., “Correlation functions in a one-dimensional Bose gas”, J. Phys. A: Math. Gen., 20 (1987), 4799–4822 | DOI | MR

[16] Ovchinnikov A. A., “Coordinate space wave function from the algebraic Bethe ansatz for the inhomogeneous six-vertex model”, Phys. Lett. A, 374 (2010), 1311–1314, arXiv: 1001.2672 | DOI | MR

[17] Kitanine N., “Correlation functions of the higher spin XXX chains”, J. Phys. A: Math. Gen., 34 (2001), 8151–8169, arXiv: math-ph/0104016 | DOI | MR | Zbl

[18] Castro-Alvaredo O. A., Maillet J. M., “Form factors of integrable Heisenberg (higher) spin chains”, J. Phys. A: Math. Theor., 40 (2007), 7451–7471, arXiv: hep-th/0702186 | DOI | MR | Zbl

[19] Deguchi T., Matsui C., “Form factors of integrable higher-spin XXZ chains and the affine quantum-group symmetry”, Nuclear Phys. B, 814 (2009), 405–438, arXiv: ; Deguchi T., Matsui C., “Correlation functions of the integrable higher-spin XXX and XXZ chains through the fusion method”, Nuclear Phys. B, 831 (2010), 359–407, arXiv: 0807.18470907.0582 | DOI | MR | Zbl | DOI | MR | Zbl

[20] Gaudin M., “Bose gas in one dimension. I. The closure property of the scattering wavefunctions”, J. Math. Phys., 12 (1971), 1674–1676 | DOI | MR

[21] Fireman E. C., Lima-Santos A., Utiel W., “Bethe ansatz solution for quantum spin-1 chains with boundary terms”, Nuclear Phys. B, 626 (2002), 435–462, arXiv: nlin.SI/0110048 | DOI | MR | Zbl

[22] Melo C. S., Martins M. J., “Algebraic Bethe ansatz for U(1) invariant integrable models: the method and general results”, Nuclear Phys. B, 806 (2009), 567–635, arXiv: 0806.2404 | DOI | MR | Zbl

[23] Belliard S., Ragoucy E., “The nested Bethe ansatz for ‘all’ closed spin chains”, J. Phys. A: Math. Theor., 41 (2008), 295202, 33 pp., arXiv: 0804.2822 | DOI | MR | Zbl

[24] Crampé N., Ragoucy E., Simon D., “Eigenvectors of open XXZ and ASEP models for a class of non-diagonal boundary conditions”, J. Stat. Mech. Theory Exp., 2010:11 (2010), P11038, 20 pp., arXiv: 1009.4119 | DOI