@incollection{statphys23_0431,
	title = {Quantum chaos in Jahn-Teller systems},
	address = {Genova, Italy},
	author = {E. Majern\'{\i}kov\'{a} and S. Shpyrko},
	booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics},
	editor = {Luciano Pietronero and Vittorio Loreto and Stefano Zapperi},
	month = {9-13 July},
	url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=431},
	year = {2007},
	biburl = {http://www.bibsonomy.org/bibtex/2f1921a260d359b6c767bfa41fdc39834/statphys23},
	abstract = {We study the statistics of the excited energy levels in a class of systems of electron-phonon interaction represented by the generalized two-level Jahn-Teller (JT) $E\otimes (b_1+b_2)$ model with quantum Hamiltonian  $\hat{H}= \Omega(b_1^{\dag}b_1+b_2^{\dag}b_2+1)I+\alpha (b_1^{\dag}+b_1)\sigma_z-\beta(b_2^{\dag}+b_2)\sigma_x$ [1] ($b_i$ are phonon (boson) operators, and $2\times 2$ Pauli matrices $\sigma_i$ account for two electron levels). The difference of electron-phonon coupling strengths $\alpha\neq\beta$ presents the generalization of the model compared to conventional JT systems with $\alpha=\beta$ and in real systems is caused, e.g., by a spatial anisotropy of a crystal. 

Nonequivalence of the phonon-electron coupling constants results in the symmetry lowering (violation of the rotation symmetry of the JT model) which is shown to have a dramatic impact on the statistics of energy levels and eigenfunctions with emerging non-conventional quantum chaotic patterns. In particular, the wavefunctions acquire multi-fractal properties and individual energy levels become extremely irregular. In spite of this the system acquires novel universalities on the level of statistical description, and these universalities appear to be irrelevant to the actual values of coupling strengths provided they differ enough one from another.
The distribution of nearest-neighbour level spacings in the domain of model parameters with mostly developed quantum chaos is shown to be close to the novel class of semi-Poisson distribution $P(S)\sim 4S\exp(-2S)$ typical,e.g. for the M-I transition in the Anderson model [2]. The vicinity to this model is also supported by the long-range statistical measure $\bar{\Delta}_3$ whose slope (spectral compressibility) appears to tend to the value (0.5/15) predicted for the said intermediate statistics [3]. Similar results emerge from the study of the fractal dimensions of the wavefunctions: the averaged over states fractal dimensions tend to a universal value in the chaotic domain in spite of extremely irregular behaviour of individual levels. The exposed results allow us to suggest that the class of investigated electron-phonon models pertains to a class of systems sharing a novel universal statistics
one of whose representatives is the Anderson chain model at the point
of M-I transition.

1)  E.Majern\'{\i}kov\'a, S.Shpyrko,  
Phys.Rev.E 73,066215 (2006);cond-mat/0509687\\
2) E.Majern\'{\i}kov\'a, S.Shpyrko,  
Phys.Rev.E 73, 057202 (2006);cond-mat/0510710\\
3) E.Majern\'{\i}kov\'a, S.Shpyrko, submitted to Phys.Rev.E; cond-mat/0611736},
	keywords = {chaos electron-phonon interaction jahn-teller multifractality quantum semi-poisson statistics statphys23 systems topic-8 wavefunctions }
}