Abstract
The relationship between one dimensional (ID) theories in field theory and Solid State Physics as well as one and two dimensional (2D) models in statistical mechanics has proved rather useful for the study of quasi-one dimensional systems. As for the exact diagonalization (of the Hamiltonian or the transfer matrix) of these models the Bethe Ansatz [1](BA) has proved to be a very powerful method. In addition to the well known solutions [2,3] this method was recently used to diagonalize the massive Thirring [4, 5] and the chiral-invariant Gross-Neveu [7, 8] models and the Kondo problem [9].
Supported in part by Brown University’s Material Research Laboratory founded by the National Science Foundation.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
H.A. Bethe, Z. Physik 71(1931)205.
E.H. Lieb and W. Liniger, Phys.Rev.130 (1963) 1605
E.H. Lieb. ibid 130, (1963)1616.
E. Brezin and J. Sinn-Justin, CR. Acad. Sci. (Paris) B263, (1966) 670.
C.N. Yang and C.P. Yang, Phys.Rev. 150 (1966) 321
C.N. Yang and C.P. Yang, Phys.Rev. 150 (1966) 327.
F.A. Berezin and V.N. Sushko,Sov. Phys. JETP 21 (1965) 865.
H. Bergnoff and H.B. Thacker, Phys.Rev.Lett. 42 (1979) 135
H. Bergnoff and H.B. Thacker, Phys.Rev. D19 (1979) 3666.
V.E. Korepin, KFKI preprint.
N. Andrei and J.H. Lowenstein, Phys.Rev.Lett. 43 (1979) 1698.
A.A. Belavin, Phys.Lett. 87B (1979) 117.
N. Andrei, Phys.Rev.Lett. 45 (1980) 379.
E. Berkcan, U. Mohanty and L. N Cooper, to be published.
F.A. Berezin and V. N. Sushko, Ref. 4 C.N. Yang, Phys. Rev. 168(1968) 1920. E. Berkcan, unpublished. V.E. Korepin, Ref.6. N.Andrei and J.H. Lowenstein, NYU preprint.
See, for example,M. Karowski, Phys.Rep. C49 (1979) 222 and the references therein.
E. Berkcan, L.N Cooper, to be published.
Since we have local relativistic theories in mind, we restrict ourselves to δ-function potentials.
C.N. Yang, Phys.Rev.Lett. 19 (1967) 1312.
For Jx ≠ Jy (≠Jz) this problem is solved by an Ansatz [21] which will hence-forth be referred to as the generalized Bethe Ansatz. This is justified by the fact that in a particular continuum limit the former reduces to the Bethe Ansatz that solves the massive Thirring model [4,5].
M. Lakshmanan, Phys. Lett. 61A (1977) 53.
V.E. Zakharov and L.A. Takhtadzhyan, Teor.Mat.Fiz 38 (1979) 26.
P.P. Kulish and E.K. Sklyanin, Phys.Lett. 70A (1979) 461. After this work has been completed it has been brought to our attention that J.H.H.Perk, Scriptie, Univ. of Amsterdam 1974 (unpublished) has also obtained the same result.
A Luther, Phys. Rev. B14 (1979) 2153.
R.J. Baxter, Phys.Rev.Lett. 26 (1971) 832
R.J. Baxter, 26(1971) 834.
J.D. Johnson, S. Krinsky and B.M. McCoy, Phys.Rev. A8 (1973) 2526.
R.J. Baxter, Ann.Israel Phys. Soc. 2(1) (1978) 37.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1981 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Berkcan, E., Cooper, L.N. (1981). Bethe Ansatz, Connection Between One-Dimensional Models and Their Classification. In: Bernasconi, J., Schneider, T. (eds) Physics in One Dimension. Springer Series in Solid-State Sciences, vol 23. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-81592-8_5
Download citation
DOI: https://doi.org/10.1007/978-3-642-81592-8_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-81594-2
Online ISBN: 978-3-642-81592-8
eBook Packages: Springer Book Archive