In mixed systems, besides regular and chaotic states, there are states supported by the chaotic region mainly living in the vicinity of the hierarchy of regular islands. We show that the fraction of these hierarchical states scales ash −α and relate the exponent α = 1 − 1/γ to the decay of the classical staying probability P (t) ∼ t −γ . This is numerically confirmed for the kicked rotor by studying the influence of hierarchical states on eigenfunction and level statistics.Typical Hamiltonian systems are neither integrable nor ergodic [1] but have a mixed phase space, where regular and chaotic regions coexist. The regular regions are organized in a hierarchical way [2] (see, e.g., Fig. 2a) and chaotic dynamics is clearly distinct from the dynamics of fully chaotic systems. In particular, chaotic trajectories are trapped in the vicinity of the hierarchy of regular islands. The most prominent quantity reflecting this, is the probability P (t) to be trapped longer than a time t, which decays as [3]in contrast to the typically exponential decay in fully chaotic systems. While the power-law decay is universal, the exponent γ is system and parameter dependent. The origin of the algebraic decay are partial transport barriers [4], e.g., Cantori, leading to a hierarchical structure of the chaotic region [5]. Quantum mechanically the classical algebraic decay of P (t) is mimicked at most until Heisenberg time [6].Even after two decades of studying quantum chaos, the search for quantum signatures of this universal power-law trapping is still in its infancy: In fact, only conductance fluctuations of open systems have been investigated so far. It was semiclassically derived that these fluctuations should have a fractal dimension D = 2 − γ/2 [7], which was confirmed in gold nanowires [8], semiconductor nanostructures [9], and numerics [10]. Quite recently, a second type of conductance fluctuations in mixed systems has been discovered numerically [11,12], namely isolated resonances. There the classical exponent γ seems to appear in the scaling of the variance of conductance increments, surprisingly, on scales below the mean level spacing, what is not understood so far. Thus, even for this subject there is a lack of basic understanding.
We study the distributions of the resonance widths P(gamma) and of delay times P(tau) in one-dimensional quasiperiodic tight-binding systems at critical conditions with one open channel. Both quantities are found to decay algebraically as gamma(-alpha) and tau(-gamma) on small and large scales, respectively. The exponents alpha and gamma are related to the fractal dimension D(E)(0) of the spectrum of the closed system as alpha = 1+D(E)(0) and gamma = 2-D(E)(0). Our results are verified for the Harper model at the metal-insulator transition and for Fibonacci lattices.
Based on a thorough numerical analysis of the spectrum of Harper's operator, which describes, e.g., an electron on a two-dimensional lattice subjected to a magnetic field perpendicular to the lattice plane, we make the following conjecture: For any value of the incommensurability parameter σ of the operator its spectrum can be covered by the bands of the spectrum for every rational approximant of σ after stretching them by factors with a common upper bound. We show that this conjecture has the following important consequences: For all irrational values of σ the spectrum is (i) a zero measure Cantor set and has (ii) a Hausdorff dimension less or equal to 1/2. We propose that our numerical approach may be a guide in finding a rigorous proof of these results.
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