Sticky eigenstates in systems with sharply-divided phase space

TL;DR

This paper explores quantum eigenstates in systems with sharply divided phase space, revealing how classical stickiness influences quantum localization and tunneling, with results applicable to non-KAM systems.

Contribution

It introduces a classification of mixed eigenstates in piecewise-linear maps and links quantum behavior to classical stickiness, extending predictions beyond KAM systems.

Findings

Sticky eigenstates scale as b1 bd in b5h in MUPO cases

Quantum tunneling contribution scales as b5h b0 b7 b5 b8 b0

Classical stickiness characterized by b5 = 2 in the studied maps

Abstract

We investigate mixed eigenstates in systems with sharply-divided phase space, using different piecewise-linear maps whose regular-chaotic boundaries are formed by marginally unstable periodic orbits (MUPOs) or by quasi-periodic orbits. With the overlap index and the entropy localization length, we classify mixed eigenstates and show that the contribution from dynamical tunneling scales as $\sim \hbar\, \exp(-b/\hbar)$, with $b>0$ associated with the relative size of the regular region. The dominant fraction of states that remain sticky to the boundaries, referred to as sticky eigenstates, scales as $\hbar^{1/2}$ in the MUPO case and oscillates around this algebraic behavior in the quasi-periodic case. This behavior generalizes established predictions for hierarchical states in KAM systems, which scale as $\hbar^{1 - 1/\gamma}$, with $\gamma$ set by the corresponding classical stickiness reflected in the algebraic decay of cumulative RTDs $t^{-\gamma}$. For the piecewise-linear maps studied here, $\gamma = 2$. These results reveal a clear quantum signature of classical stickiness in non-KAM systems.