[Paper Review] Quantum chaos, integrability, and late times in the Krylov basis

Vijay Balasubramanian, Javier M. Magán+1 more|arXiv (Cornell University)|Dec 6, 2023

Stochastic processes and statistical mechanics4 citations

TL;DR

This paper proposes that quantum chaotic systems are characterized by universal, Random Matrix Theory (RMT)-like statistics in the local mean and covariance of their Lanczos spectrum—derived from tridiagonalizing the Hamiltonian in the Krylov basis. It demonstrates that these universal statistics fully determine late-time dynamics, including spectral form factors and spread complexity, and analytically computes long-time averaged Krylov state probabilities, revealing a connection between eigenstate complexity and universality classes.

ABSTRACT

Quantum chaotic systems are conjectured to display a spectrum whose fine-grained features (gaps and correlations) are well described by Random Matrix Theory (RMT). We propose and develop a complementary version of this conjecture: quantum chaotic systems display a Lanczos spectrum whose local means and covariances are well described by RMT. To support this proposal, we first demonstrate its validity in examples of chaotic and integrable systems. We then show that for Haar-random initial states in RMTs the mean and covariance of the Lanczos spectrum suffices to produce the full long time behavior of general survival probabilities including the spectral form factor, as well as the spread complexity. In addition, for initial states with continuous overlap with energy eigenstates, we analytically find the long time averages of the probabilities of Krylov basis elements in terms of the mean Lanczos spectrum. This analysis suggests a notion of eigenstate complexity, the statistics of which differentiate integrable systems and classes of quantum chaos. Finally, we clarify the relation between spread complexity and the universality classes of RMT by exploring various values of the Dyson index and Poisson distributed spectra.

Motivation & Objective

• To establish a new characterization of quantum chaos based on universal fine-grained statistics of the Lanczos spectrum, complementing traditional energy-level statistics.
• To demonstrate that the local mean and covariance of the Lanczos spectrum—when matching RMT—fully control long-time dynamics such as survival amplitudes and spread complexity.
• To derive analytical expressions for long-time averaged Krylov basis state probabilities for initial states with continuous energy overlap.
• To clarify the relationship between spread complexity and RMT universality classes, including Dyson index dependence and Poisson-like spectra.
• To connect the Lanczos tail behavior to black hole information paradox and non-perturbative quantum gravity effects like wormholes.

Proposed method

• Tridiagonalize the Hamiltonian using the Lanczos algorithm in the Krylov basis generated by a fixed initial state.
• Analyze the Lanczos coefficients (the Lanczos spectrum) for chaotic and integrable systems, comparing their local mean and covariance to RMT predictions.
• Use RMT ensembles with varying Dyson indices (β = 1, 2, 4) and Poisson-distributed spectra to test universality of Lanczos statistics.
• Derive an analytical formula for the long-time average of Krylov basis state probabilities using the mean Lanczos spectrum.
• Relate the decay of the Lanczos tail to late-time saturation of correlation functions and complexity, linking to black hole physics and non-perturbative gravity effects.
• Apply the framework to continuum systems by restricting to microcanonical or matrix model regimes for applicability to quantum gravity.

Experimental results

Research questions

• RQ1 Can the universal statistics of the Lanczos spectrum—specifically its local mean and covariance—characterize quantum chaos independently of energy-level statistics?
• RQ2 To what extent do the Lanczos spectrum's universal features predict the long-time behavior of survival amplitudes and spectral form factors in chaotic systems?
• RQ3 How do the long-time averaged probabilities of Krylov basis states depend on the mean Lanczos spectrum for initial states with continuous energy support?
• RQ4 What is the role of the Dyson index and Poisson-distributed spectra in shaping the slope of the spread complexity function?
• RQ5 How does the decay of the Lanczos tail at the end of the Krylov chain explain the non-decaying plateau in correlation functions at late times, relevant to the black hole information paradox?

Key findings

• The Lanczos spectrum of quantum chaotic systems exhibits universal local mean and covariance statistics matching those of Random Matrix Theory (RMT), even when energy-level statistics are not fully RMT-like.
• For Haar-random initial states in RMT, the mean and covariance of the Lanczos spectrum fully determine the long-time behavior of the spectral form factor and spread complexity.
• The long-time averaged probabilities of Krylov basis states are analytically computed in terms of the mean Lanczos spectrum for initial states with continuous energy overlap, revealing a universal steady-state distribution.
• The statistics of eigenstate complexity, derived from the Lanczos spectrum, differentiate integrable systems from chaotic ones, with distinct universal behaviors.
• The decay of the Lanczos tail at the end of the Krylov chain controls the late-time saturation of correlation functions, explaining the plateau behavior in finite-entropy chaotic systems.
• The framework connects the Lanczos tail to non-perturbative quantum gravity effects such as higher-genus wormholes, suggesting a dynamical mechanism for information retention in black holes.