Fractional-Power-Law Level Statistics Due to Dynamical Tunneling

Abstract

For systems with a mixed phase space we demonstrate that dynamical tunneling universally leads to a fractional power law of the level-spacing distribution P(s) over a wide range of small spacings s. Going beyond Berry-Robnik statistics, we take into account that dynamical tunneling rates between the regular and the chaotic region vary over many orders of magnitude. This results in a prediction of P(s) which excellently describes the spectral data of the standard map. Moreover, we show that the power-law exponent is proportional to the effective Planck constant h(eff).

Consequences of flooding on spectral statistics

“…III D we unify this prediction with the results of Ref. [31], leading to the more sophisticated flooding-and tunneling-improved Berry-Robnik distribution P fti (s). This distribution additionally accounts for the effects of level repulsion between regular and chaotic states.…”

“…The derivation is done along the lines of Ref. [31]. We incorporate the effects of flooding into this theory by replacing the number of regular states N sc r with the number of surviving regular states N f r which fulfill the existence criterion (20).…”

“…[7,28,30,31] the flooding-and tunnelingimproved Berry-Robnik distribution P fti (s) consists of three distinct parts: P fti (s) = p r-r (s) + p c-c (s) + p r-c (s).…”

“…If such tunneling couplings are small, regular eigenstates will typically have tiny chaotic admixtures and vice versa. The influence of such weak couplings on spacing statistics can be described perturbatively [27][28][29][30][31]. Based on this description a tunneling-improved Berry-Robnik distribution was de- rived recently, which explains the power-law distribution of small spacings in mixed systems [31].…”

Exact treatment of planar two-electron quantum dots: Effects of anharmonicity on the complexity

“…A remedy is found by extending the idea underlying the Berry-Robnik distribution, which is to split the classical phase space into distinct regular and chaotic regimes. The effects of dynamical tunneling 42 and chaos assisted tunneling 43 connect these two classically distinct regimes in the quantum regime and already improve the results 44,45 . Recent additional achievements considering the effects of flooding 46 appear to complete the discussion, at least for the nextneighbor spacing distribution.…”