Spectral Edge Rigidity of Quantum Chaotic States

Abstract

We determine the distribution of fidelity susceptibility for chaotic eigenstates at the spectral edge of Gaussian random-matrix ensembles. Previous work showed that, in the unitary class, the characteristic susceptibility scale of edge states grows as $D^{1/3}$, rather than proportionally to $D$ as in the spectral bulk, reflecting Airy-edge level rigidity. Extending a determinant-based framework introduced for bulk states, we derive the universal edge distributions for both the orthogonal and unitary ensembles. The two symmetry classes share the scaling variable $g/D^{1/3}$ and exhibit a symmetry-dependent cubic suppression of small susceptibilities, while their algebraic large-$g$ tails reflect the corresponding symmetry-dependent level repulsion. Although eigenvector statistics retain their random-matrix form throughout the spectrum, edge rigidity makes low-lying chaotic states parametrically less sensitive to generic perturbations than bulk states. Our results establish universal, symmetry-dependent spectral-edge fidelity-susceptibility statistics in systems whose chaotic dynamics extends down to the ground state.