Universal Relation between Spectral and Wavefunction Properties at Criticality

Abstract

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 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 exist 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 energy spectrum, such as spectral compressibility $\chi$, 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 non-local 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 banded random matrices as representatives of the second class. Our thorough numerical analysis supports validity of the simple relation $\chi +D_1=1$ 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 $D_1(r)$, where $r$ is the averaged level spacing ratio, which is valid for a broad class of critical systems.

11 pages, 7 figures

References in corpus (30)

• From Quantum Chaos and Eigenstate Thermalization to Statistical Mechanics and Thermodynamics
• Anderson Transitions
• Localization of interacting fermions at high temperature
• The distribution of the ratio of consecutive level spacings in random matrix ensembles
• Quantum chaos challenges many-body localization
• Random Matrices and Chaos in Nuclear Physics
• Multifractal Scalings across the Many-Body Localization Transition
• Fractal superconductivity near localization threshold
• Ergodicity Breaking Transition in Finite Disordered Spin Chains
• Anderson localization on random regular graphs
• Polynomially filtered exact diagonalization approach to many-body localization
• Non-ergodic phases in strongly disordered random regular graphs
• Many-Body Localization in the Age of Classical Computing
• Multifractality at the quantum Hall transition: Beyond the parabolic paradigm
• Critical properties of the Anderson localization transition and the high dimensional limit
• Eigenfunction entropy and spectral compressibility for critical random matrix ensembles
• The integer quantum Hall plateau transition is a current algebra after all
• Dimensional Dependence of Critical Exponent of the Anderson Transition in the Orthogonal Universality Class
• Renormalization Group Analysis of the Anderson Model on Random Regular Graphs
• Efficient algorithm to compute the second Chern number in four dimensional systems
• Scale-Invariant Survival Probability at Eigenstate Transitions
• Scale-invariant critical dynamics at eigenstate transitions
• Localization challenges quantum chaos in the finite two-dimensional Anderson model
• Level compressibility in a critical random matrix ensemble: The second virial coefficient
• Random Cantor sets and mini-bands in local spectrum of quantum systems
• Estimate of the critical exponent of the Anderson transition in the three and four dimensional unitary universality classes
• Spectral statistics of Toeplitz matrices
• Statistical properties of structured random matrices
• Renormalization group for Anderson localization on high-dimensional lattices
• Level number variance and spectral compressibility in a critical two-dimensional random matrix model