Universal relation between spectral and wavefunction properties at criticality

Quantum-chaotic systems exhibit several universal properties, ranging from level repulsion in the energy spectrum to wavefunction delocalization. On the other hand, if wavefunctions are localized, the energy levels exhibit no level repulsion and their statistics is Poisson. At the boundary between quantum chaos and localization, however, one observes critical behavior, not complying with any of those characteristics. An outstanding open question is whether there exists yet another type of universality, which is genuine for the critical point. Previous work suggested that there may exist a relation between the global characteristics of the energy spectrum, such as the spectral compressibility χ , and the degree of wavefunction delocalization, expressed via the fractal dimension D 1 of the Shannon–von Neumann entropy in a preferred (e.g., real-space) basis. Here, we study physical systems subject to local and nonlocal hopping, both with and without time-reversal symmetry, with the Anderson models in dimensions three to five being representatives of the first class, and the random banded matrices as representatives of the second class. Our thorough numerical analysis supports the validity of the simple relation χ + D 1 = 1 in all systems under investigation. Hence, we conjecture that it represents a universal property of a broad class of critical models. Moreover, we test and confirm the accuracy of our surmise for a closed-form expression of the spectral compressibility in the one-parameter critical manifold of random banded matrices. Based on these findings, we derive a universal function D 1 ( r ) of the averaged level spacing ratio r , which is valid for a broad class of critical systems.