Skip to Main Content

Access Provided by Xinjiang University

Physical Review E

Access by Xinjiang University

Quantum chaotic patterns in the $E \otimes (b_{1} + b_{2})$ Jahn-Teller model

Eva Majerníková^1,2,* and S. Shpyrko^2,†

¹Institute of Physics, Slovak Academy of Sciences, Dúbravská cesta 9, SK-84 511 Bratislava, Slovak Republic
²Department of Theoretical Physics, Palacký University, Tř. 17. listopadu 50, CZ-77207 Olomouc, Czech Republic

^*Electronic address: fyziemar@savba.sk
^†On leave from the Institute of Nuclear Research, Ukrainian Academy of Sciences, pr. Nauki 47 Kiev, Ukraine.

Phys. Rev. E 73, 057202 – Published 24 May, 2006

DOI: https://doi.org/10.1103/PhysRevE.73.057202

Abstract

We study statistical properties of excited levels of the $E \otimes (b_{1} + b_{2})$ Jahn-Teller model. The multitude of avoided crossings of energy levels is generally claimed to be a testimony of quantum chaos. We found that apart from two limiting cases ( $E \otimes e$ and Holstein model) the distribution of nearest-neighbor spacings is rather stable as to the change of parameters and different from the Wigner one. This limiting distribution assumably shows scaling $\sim \sqrt{S}$ at small $S$ and resembles the semi-Poisson law $P (S) = 4 S \exp (- 2 S)$ at $S \geq 1$ . The latter is believed to be universal and characteristic, e.g., at the transition between metal and insulator phases.

Article Text

References (29)

H. Eiermann and M. Wagner, J. Chem. Phys. 96, 4509 (1992).
A. Köngeter and M. Wagner, J. Chem. Phys. 92, 4003 (1990).
R. Graham and M. Höhnerbach, Z. Phys. B: Condens. Matter 57, 233 (1984); Phys. Lett. 101A, 61 (1984).
J. Eidson and R. F. Fox, Phys. Rev. A 34, 3288 (1986); R. F. Fox and J. Eidson, 34, 482 (1986); 36, 4321 (1987).
R. Blumel and B. Esser, Phys. Rev. Lett. 72, 3658 (1994); B. Esser and H. Schanz, Z. Phys. B: Condens. Matter 96, 553 (1995); H. Schanz and B. Esser, ibid. 101, 299 (1996); Phys. Rev. A 55, 3375 (1997).
M. Sonnek et al., Phys. Rev. B 49, 15637 (1994); U. Herfort and M. Wagner, J. Phys.: Condens. Matter 13, 3297 (2001).
U. Herfort and M. Wagner, J. Phys.: Condens. Matter 13, 3297 (2001).
B. Eckhardt, Phys. Rep. 163, 205 (1988).
K. Nakamura, Quantum Chaos—A New Paradigm of Nonlinear Dynamics (Cambridge University Press, Cambridge, 1993).
T. A. Brody et al., Rev. Mod. Phys. 53, 385 (1981).
H. B. Shore and L. M. Sander, Phys. Rev. B 7, 4537 (1973).
E. Majerníková et al., Phys. Rev. B 65, 174305 (2002).
Y. E. Perlin and M. Wagner, in The Dynamical Jahn-Teller Effect in Localized Systems, edited by Y. Perlin and M. Wagner (Elsevier Science Publishers, New York, 1984), p. 1.
E. Majerníková and S. Shpyrko, J. Phys.: Condens. Matter 15, 2137 (2003).
H. Yamasaki et al., Phys. Rev. E 68, 046201 (2003).
M. C. M. O'Brien, Proc. R. Soc. London, Ser. A A281, 323 (1964).
E. Majerníková and S. Shpyrko, Phys. Rev. E (to be published); cond-mat/0509687.
H. C. Longuet-Higgins et al., Proc. R. Soc. London, Ser. A A244, 1 (1958).
K. Takahashi, Prog. Theor. Phys. Suppl. 98, 109 (1989).
M. L. Mehta, Nucl. Phys. 18, 395 (1960); M. L. Mehta and M. Gaudin, ibid. 18, 420 (1960); M. Gaudin, ibid. 25, 447 (1961).
P. Gaspard et al., Phys. Rev. A 42, 4015 (1990).
N. Rosenzweig and C. E. Porter, Phys. Rev. 120, 1968 (1960).
D. M. Leitner, Phys. Rev. E 48, 2536 (1993).
D. M. Leitner et al., Phys. Rev. Lett. 73, 2970 (1994); E. Haller et al., 52, 1665 (1984).
M. Robnik, J. Phys. A 20, L495 (1987).
H. Hasegawa et al., Phys. Rev. A 38, 395 (1988).
S. N. Evangelou and J.-L. Pichard, Phys. Rev. Lett. 84, 1643 (2000); S. N. Evangelou and D. E. Katsanos, Phys. Lett. A 334, 331 (2005); E. B. Bogomolny et al., Phys. Rev. E 59, R1315 (1999).
B. I. Shklovskii et al., Phys. Rev. B 47, 11487 (1993).
H.-D. Meyer et al., J. Phys. A 17, L831 (1984); A. Shudo, Prog. Theor. Phys. Suppl. 98, 173 (1989).