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Parabolic maps with spin: Generic spectral statistics with non-mixing classical limit

Grischa Haag·Stefan Keppeler
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Abstract

We investigate quantised maps of the torus whose classical analogues are ergodic but not mixing. Their quantum spectral statistics shows non-generic behaviour, i.e.it does not follow random matrix theory (RMT). By coupling the map to a spin 1/2, which corresponds to changing the quantisation without altering the classical limit of the dynamics on the torus, we numerically observe a transition to RMT statistics. The results are interpreted in terms of semiclassical trace formulae for the maps with and without spin respectively. We thus have constructed quantum systems with non-mixing classical limit which show generic (i.e. RMT) spectral statistics. We also discuss the analogous situation for an almost integrable map, where we compare to Semi-Poissonian statistics.

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Grischa Haag, Stefan Keppeler. 2001-06-27. Parabolic maps with spin: Generic spectral statistics with non-mixing classical limit. https://doi.org/10.1088/0951-7715%2F15%2F1%2F304

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