Scars on Quantum Networks Ignore the Lyapunov Exponent

We show that enhanced wave function localization due to the presence of short unstable orbits and strong scarring can rely on completely different mechanisms. Specifically we find that in quantum networks the shortest and most stable orbits do not support visible scars, although they are responsible for enhanced localization in the majority of the eigenstates. Scarring orbits are selected by a criterion which does not involve the classical stability. We obtain predictions for the energies of visible scars and the distributions of scarring strengths and inverse participation numbers.

Figure 1

Statistical properties of eigenstates for fully connected graphs with different valencies $v=V-1$. The eigenstates were obtained by diagonalization of the bond-scattering matrix, Eq. (4), and a statistical ensemble was generated by choosing random bond lengths. (a) The bond intensities of two representative eigenstates are shown with a gray scale for a typical state (left) and a scar on a triangular PO (right). (b) Probability distribution of the inverse participation numbers (IPN), showing a steplike cutoff at $\mathcal{I}=1/6$ that can be attributed to scarring on triangular orbits. (c) The bulk of the IPN distribution shows scaling according to Eq. (2). (d) The quantum return probability and a classical approximation ($*$) based only on the period-two orbits. (e) Rescaled and integrated distribution of triangle scars ${\mathcal{P}}_{\mathrm{i}\mathrm{n}\mathrm{t}}(\delta )=B^{-1}{\int }_0^{\delta }d{\delta }^{\prime }{\mathcal{P}}^{(3)}({\delta }^{\prime })$. The dashed line has slope 1, corresponding to the theoretical prediction, Eq. (12).

Figure 2

$\mathcal{P}(\mathcal{I})$ for star-graphs of various valencies (inset a star-graph with $v=6$). In the upper inset we report the integrated distribution ${\mathcal{P}}_{\mathrm{i}\mathrm{n}\mathrm{t}}(\delta )={\int }_0^{\delta }d{\delta }^{\prime }P({\delta }^{\prime })$ in a double-logarithmic plot. The dashed line has slope 0.5 which corresponds to the theoretical prediction Eq. (12).