Krylov fractality and complexity in generic random matrix ensembles

Abstract

Krylov space methods provide an efficient framework for analyzing the dynamical aspects of quantum systems, with tridiagonal matrices playing a key role. Despite their importance, the behavior of such matrices from chaotic to integrable states, transitioning through an intermediate phase, remains unexplored. We aim to fill this gap by considering the properties of the tridiagonal matrix elements and the associated basis vectors for appropriate random matrix ensembles. We utilize the Rosenzweig-Porter model as our primary example, which hosts a fractal regime in addition to the ergodic and localized phases. We discuss the characteristics of the matrix elements and basis vectors across the three (ergodic, fractal, and localized) regimes and introduce tools to identify the transition points. The exact expressions of the Lanczos coefficients are provided in terms of $q$-logarithmic function across the full parameter regime. The numerical results are corroborated with analytical reasoning for certain features of the Krylov spectra. Additionally, we investigate the Krylov state complexity within these regimes, showcasing the efficacy of our methods in pinpointing these transitions.

v2: minor changes, typos corrected, to appear in Phys. Rev. B (Letter)

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Cited by in corpus (15)

• Krylov space approach to Singular Value Decomposition in non-Hermitian systems
• Krylov Complexity in Mixed Phase Space
• Brickwall One-Loop Determinant: Spectral Statistics & Krylov Complexity
• Krylov Complexity as a Probe for Chaos
• Singular Value Decomposition and Its Blind Spot for Quantum Chaos in Non-Hermitian Sachdev-Ye-Kitaev Models
• Diagnosing Quantum Many-body Chaos in Non-Hermitian Quantum Spin Chain via Krylov Complexity
• Properties of Krylov state complexity in qubit dynamics
• Krylov Complexity Meets Confinement
• Free Probability approach to spectral and operator statistics in Rosenzweig-Porter random matrix ensembles
• Sparsity dependence of Krylov state complexity in the SYK model
• Krylov-space anatomy and spread complexity of a disordered quantum spin chain
• Complexity of Quadratic Quantum Chaos
• Quantum Complexity in Rule-Based Constrained Many-Body Models: Scars, Fragmentation, and Chaos
• Krylov Complexity Under Hamiltonian Deformations and Toda Flows
• Higher-Order Krylov State Complexity in Random Matrix Quenches