Quantum chaos, localized states and clustering in excited spectra of Jahn-Teller models

Abstract

We studied complex spectra of spin-two boson systems represented by E$\otimes$e and E$\otimes (b_1+b_2)$ Jahn-Teller models. For E$\otimes$e, at particular rotation quantum numbers we found a coexistence of up to three regions of the spectra, (i) the dimerized region of long-range ordered (extended) pairs of oscillating levels, (ii) the short-range ordered (localized) "kink lattice" of avoiding levels, and (iii) the intermediate region of kink nucleation with variable range of ordering. This structure appears above certain critical line as a function of interaction strength. The level clustering and level avoiding generic patterns reflect themselves in several intermittent regions between up-to three branches of spectral entropies. Linear scaling behavior of the widths of curvature probability distributions provides the conventionally adopted indication for the presence of quantum chaos. The mapping onto classical integrable Calogero-Moser gas provided useful insight into the complex level dynamics, including the soliton collisions representing the level avoidings and, in a range of model parameters, a novel view on the notion of quantum chaos formulated in terms of quantum numbers via the logistic equation. We found that apart from two limiting cases of $E\otimes(b_1+b_2)$ model ($E\otimes e$ and Holstein model) the distribution of nearest neighbor spacings of this model is rather stable as to the change of parameters and different from Wigner one. This limiting distribution assumably shows scaling $\sim\sqrt{S}$ at small $S$ and resembles the semi-Poisson law $P(S)= 4S \exp (-2 S)$ at $S\geq 1$. The latter is believed to be universal and characteristic, e.g. at the transition between metal and insulator phases.