Abstract
It is shown how the Bethe Ansatz (BA) analysis for the quantum statistical mechanics of the Nonlinear Schrodinger Model generalises to the other quantum integrable models and to the classical statistical mechanics of the classical integrable models. The bose-fermi equivalence of these models plays a fundamental role even at classical level. Two methods for calculating the quantum or classical free energies are developed: one generalises the BA method the other uses functional integral methods. The familiar classical action-angle variables of the integrable models developed for the real line R are used throughout, but the crucial importance of periodic boundary conditions is recognized and these are imposed. Connections with the quantum inverse method for quantum integrable systems are established. The R-matrix and the Yang-Baxter relation play a fundamental role in the theory. The lectures draw together the quantum BA method, the quantum inverse method, and the generalised BA and functional integral methods introduced more recently.
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
C. N. Yang, C. P. Yang, J. Math. Phys. 10 (1969) 1115.
H. B. Thacker, Rev. Mod. Phys. 53 (1981) 253.
R. K. Bullough, P. J. Caudrey, in Solitons, ed. R. K. Bullough and P. J. Caudrey (Springer, Heidelberg, 1980) and the other chapters there.
E. K. Sklyanin, LOMI preprint E-3–1979 (1979).
R. K. Bullough, in Nonlinear Phenomena in Physics, ed. F. Claro (Springer, Heidelberg, 1985), pp. 70–102; R. K. Bullough, D. J. Pilling, J. Timonen, in Nonlinear Phenomena in Physics, pp. 103–128.
M. Lakshmanan, Phys. Lett. 61A (1977) 53.
L. A. Takhtadzhyan, Phys. Lett. 64A (1977) 235.
S. Coleman, Phys. Rev. D11 (1975) 2088.
M. Takahashi, M. Suzuki, Prog. Theor. Phys. 48 (1972) 2187.
H. Bergnoff, H. B. Thacker, Phys. Rev. D19 (1979) 3666.
Korepin [12,13] in effect uses the different relation between μ and g0 that \( \mu = \frac{1}{2}\left( {\pi + {g_o}} \right) \). In [12] he treats the attractive MTM with 0 < g0 < π (π/2<μ<π or, since \( \pi - \mu = \frac{1}{8}\;{\gamma_o} \) (see below), 4π > γ0 > 0) and in [13] he treats a repulsive MTM with -π < g0 < 0 (0 < μ < π/2 or 8π > γ0 > 4π).
V. E. Korepin, TMP (USSR) 41 (1979) 169.
V. E. Korepin, Comm. Math. Phys. 76 (1980) 165.
A. Luther, Chap. 12 in Solitons Ref. [3], pp. 355–372.
M. Jimbo, T. Miwa, Y. Mori, M. Sato, Physica 1D (1980) 80 and references.
E. K. Sklyanin, L. A. Takhtadzhyan, L. D. Faddeev, Theor. Mat. 40 (1979) 194.
R. F. Dashen, B. Hasslacher, A. Neveu, Phys. Rev. D11 (1975) 3424.
R. K. Bullough, D. J. Pilling, J. Timonen, J. Phys. A: Math. Gen. 19 (1986) L955.
R. K. Bullough, D. J. Pilling, J. Timonen, in Physics of Many-Particle Systems, ed. A. S. Davydov (Ukrainian Academy of Sciences of the USSR, Kiev, 1986).
P. Goddard, D. Olive, Int. J. Mod. Phys. A1, No. 2 (1986) 303.
M. Jimbo, T. Miwa, in Vertex Operators in Mathematical Physics, ed. J. Lepowsky, S. Mandelstam and I. M. Singer (Springer, Heidelberg, 1984), pp. 275–290 and other papers there.
V. Kac, Infinite Dimensional Lie Algebras—An Introduction, 2d ed (Cambridge University Press, Cambridge, 1985).
For example S. Olafsson, R. K. Bullough, to be published.
P. P. Kulish, E. K. Sklyanin, “Quantum Spectral Transform Method: Recent Developments”, in Proc. of the Tvärminne Symposium, Finland, 1981, ed. J. Hietarinta and C. Montonen (Springer, Heidelberg, 1982).
L. D. Faddeev, in Proc. Ecole d’Eté de Physique Theorique, Les Houches 1982, ed. R. Stora and J. B. Zuber (North-Holland, Amsterdam, 1983).
M. Wadati and Y. Akutsu, Exactly Solvable Models in Statistical Mechanics, in this volume.
H. J. de Vega, in Proc. Symp. on Topological and Geometrical Methods in Field Theory (World Scientific, Singapore, 1986), in press.
V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer, Heidelberg, 1978).
R. K. Dodd, R. K. Bullough, Physica Scr. 20 (1979) 514.
L. D. Faddeev, Chap. 11 in Solitons Ref. [3], pp. 339–354.
D. J. Scalapino, M. Sears and R. S. Ferrell, Phys. Rev. B6 (1972) 3409.
E. H. Lieb and W. Liniger, Phys. Rev. 130 (1963) 1605.
A. G. Izergin and V. E. Korepin, Lett. Math. Phys. 5 (1981) 199.
A. G. Izergin and V. E. Korepin, in Problems in Quantum Field Theory and Statistical Physics 3, ed. P. P. Kulish and V. N. Popov, LOMI Vol. 120 “Nauka” Leningrad.
Yi Cheng, DUNG Ph.D. Thesis, University of Manchester (1987).
J. Timonen, M. Stirland, D. J. Pilling, Yi Cheng and R. K. Bullough, Phys. Rev. Lett. 56 (1986) 2233.
J. T. Timonen, R. K. Bullough and D. J. Pilling, Phys. Rev. B34 (1986) 6525.
J. T. Timonen, R. K. Bullough and D. J. Pilling, Classical Limit of Bethe Ansatz Statistical Mechanics for the Massive Thirring Model, to be published.
V. E. Korepin, in Completely Solvable Systems in Field Theory, Lecture Notes at SERC-LMS Symposium, University of Durham (1986).
D. J. Pilling, R. K. Bullough and J. Timonen, to be published.
C. Itzykson and J. B. Zuber, Quantum Field Theory (McGraw-Hill, New York, 1980)
J. Timonen, D. J. Pilling and R. K. Bullough, in Coherence, Cooperation and Fluctuations, ed. F. Haake, L. M. Narducci and D. F. Walls (CUP, Cambridge, 1986), pp. 18–34. Unfortunately, in this paper a copying error replacing k = h(k͂) by k͂ = h(k) introduced sign errors in the equivalents of our equations (5.9) and (5.22).
M. Wadati, J. Phys. Soc. Japan 54 (1985) 3727.
R. K. Bullough, D. J. Pilling and J. Timonen, in Magnetic Excitations and Fluctuations, ed. S. W. Lovesey, U. Balucani, F. Borsa, V. Tognetti (Springer, Berlin, 1984), pp. 80–85.
R. K. Bullough, D. J. Pilling and J. Timonen, in Dynamical Problems in Soliton Systems, ed. S. Takeno (Springer, Heidelberg, 1985), pp. 105–114.
S. G. Chung, Y. C. Chang, Phys. Rev. Lett. 50 (1983) 791.
N. N. Chen, M. D. Johnson and M. Fowler, Phys. Rev. Lett. 56 (1986) 907;
N. N. Chen, M. D. Johnson and M. Fowler, Phys. Rev. Lett. 56 (1986) 1427 (Erratum).
Yu- zhong Chen, D. J. Pilling, R. K. Bullough and J. Timonen, to be published.
R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill Book Co., New York, 1965).
Yu-zhong Chen, D. J. Pilling, R. K. Bullough and J. Timonen, to be published.
M. Karowski, H. J. Thun, J. T. Truong and P. H. Weiss, Phys. Lett. 67B (1977) 321.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1988 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bullough, R.K., Pilling, D.J., Timonen, J. (1988). Soliton Statistical Mechanics: Statistical Mechanics of the Quantum and Classical Integrable Models. In: Lakshmanan, M. (eds) Solitons. Springer Series in Nonlinear Dynamics . Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-73193-8_18
Download citation
DOI: https://doi.org/10.1007/978-3-642-73193-8_18
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-73195-2
Online ISBN: 978-3-642-73193-8
eBook Packages: Springer Book Archive