# Universal relation between spectral and wavefunction properties at criticality

## TL;DR

The paper discovers a universal relationship between energy spectrum and wavefunction properties in quantum systems at criticality.

## Contribution

A new universal relation, χ + D1 = 1, is conjectured and numerically confirmed for critical systems.

## Key findings

- The relation χ + D1 = 1 holds across various critical systems with different symmetries and dimensions.
- A universal function D1(r) is derived based on the averaged level spacing ratio r for critical systems.
- The findings suggest a broader universality at criticality beyond quantum chaos and localization.

## Abstract

An important role in physics research is to uncover universal properties of various systems with different microscopic descriptions. Examples of microscopic models that exhibit paradigmatic properties are those that describe chaotic quantum dynamics and have spectral and wavefunction properties governed by random-matrix theory. Radical counterexamples to this behavior are also known, and one of such cases is the well-known Anderson localization. Nevertheless, much less is known about the possible universal properties at the boundary between quantum chaos and localization. Here, we conjecture and confirm, using large-scale numerical simulations, a universal relation between the spectral compressibility and the wavefunctions’ fractal dimension at criticality. This result paves way toward searching analogous relations in interacting models.

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 D1 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+D1=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 D1(r) of the averaged level spacing ratio r, which is valid for a broad class of critical systems.