Abstract
We study two-magnon Bethe states in the spin-1/2 XXZ chain. The string hypothesis assumes that complex rapidities of the bound states take special forms. It is known, however, that there exist “non-string states,” which substantially disagrees with the string hypothesis. In order to clarify their nature, we study the large-N behavior of solutions of the Bethe-Ansatz equations to obtain explicit forms of typical Bethe states, where N is the length of the chain, and apply the scaling analysis (the multifractal analysis) to the Bethe states. It turns out that the non-string states contain “quasi-bound” states, which in some sense continuously interpolate between extended states and localized states. The “quasi-bound” states can be distinguished from known three types of states, i.e., extended, localized, and critical states. Our results indicate that there might be a need to reconsider the standard classification scheme of wavefunctions.
Similar content being viewed by others
REFERENCES
H. Bethe, Z. Phys. 71:205, (1931).
R. Orbach, Phys. Rev. 112:309, (1958).
M. Takahashi, Prog. Theor. Phys. 46:401 (1971); M. Gaudin, Phys. Rev. Lett. 26:1301 (1971); M. Takahashi, and M. Suzuki, Prog. Theor. Phys. 46:2187 (1972).
F. Woynarovich, J. Phys. A15:2985, (1982); C. Destri, and J. H. Lowenstein, Nucl. Phys. B 205[FS5]:369 (1982); O. Babelon, H. J. de Vega, and C. M. Viallet, Nucl. Phys. B 220[FS8]:13, (1983).
A. A. Vladimirov, Phys. Lett. 105A:418 (1984).
F. H. L. Essler, V. E. Korepin, and K. Schoutens, J. Phys. A 25:4115 (1992); K. Isler, and M. B. Paranjape, Phys. Lett. 319B:209 (1993); G. Jüttner and B. D. Dörfel, J. Phys. A 26:3105 (1993).
See, for a review, H. Hiramoto, and M. Kohmoto, Int. J. Mod. Phys. B 6:281 (1992).
M. Kohmoto, L. P. Kadanoff, and C. Tang, Phys. Rev. Lett. 50:1870 (1983).
S. Ostlund, R. Pandit, D. Rand, H. J. Schellhuber, and E. D. Siggia, Phys. Rev. Lett. 50:1873 (1983).
S. Katsura, Ann. Phys. 31:325 (1965).
T. Koma, and H. Ezawa, Prog. Theor. Phys. 78:1009 (1987).
F. C. Alcaraz, M. N. Barber, and M. T. Batchelor, Phys. Rev. Lett. 58:771 (1987).
T. C. Halsey, M. H. Jensen, L. Kadanoff, I. Procaccia, and B. Shraiman, Phys. Rev. A 33:1141 (1986).
See, for example, M. Reed, and B. Simon, Methods of Modern Mathematical Physics I, Functional Analysis, (Academic Press, New York, 1972), Chapter VII.
B. Huckstein, and L. Schweitzer, Phys. Rev. Lett. 72:713 (1994).
R. B. Griffiths, J. Math. Phys. 5:1215 (1964).
R. B. Griffiths, in Phase Transition and Critical Phenomena, Vol. 1, C. Domb and M. S. Green, eds., (Academic Press, New York, 1972).
Y. Kato, N. Mugibayashi, and K. Sekine, in RIMS Kôkyûroku No. 159, 1972, The Hamiltonian, its definition and spectrum, p. 69, (Research Institute for Mathematical Sciences, Kyoto University, Kyoto), [in Japanese].
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Morita, Y., Kohmoto, M. & Koma, T. Quasi-Bound States of Two Magnons in the Spin-1/2XXZ Chain. Journal of Statistical Physics 88, 745–780 (1997). https://doi.org/10.1023/B:JOSS.0000015170.66153.4d
Issue Date:
DOI: https://doi.org/10.1023/B:JOSS.0000015170.66153.4d