Quantum signatures of classical multifractal measures

TL;DR

This paper explores how multifractal classical measures influence the quantum Weyl law, revealing that the governing dimension differs from the trivial classical dimension and oscillates with system size, supported by numerical simulations and a classical model.

Contribution

It demonstrates that the quantum Weyl law is affected by multifractality, introducing a non-trivial asymptotic dimension different from the classical invariant set dimension.

Findings

Weyl law governed by a dimension different from D_0

Observed dimension oscillates with system size M

Classical model explains multifractal effects on Weyl law

Abstract

A clear signature of classical chaoticity in the quantum regime is the fractal Weyl law, which connects the density of eigenstates to the dimension $D_0$ of the classical invariant set of open systems. Quantum systems of interest are often {\it partially} open (e.g., cavities in which trajectories are partially reflected/absorbed). In the corresponding classical systems $D_0$ is trivial (equal to the phase-space dimension), and the fractality is manifested in the (multifractal) spectrum of R\'enyi dimensions $D_q$. In this paper we investigate the effect of such multifractality on the Weyl law. Our numerical simulations in area-preserving maps show for a wide range of configurations and system sizes $M$ that (i) the Weyl law is governed by a dimension different from $D_0=2$ and (ii) the observed dimension oscillates as a function of $M$ and other relevant parameters. We propose a classical model which considers an undersampled measure of the chaotic invariant set, explains our two observations, and predicts that the Weyl law is governed by a non-trivial dimension $D_\mathrm{asymptotic} < D_0$ in the semi-classical limit $M\rightarrow\infty$.