Quantum chaos, integrability, and late times in the Krylov basis

TL;DR

The paper proposes a complementary chaos diagnostic: chaotic quantum systems possess a Lanczos spectrum whose local mean and covariances align with Random Matrix Theory. By deriving and validating mean/covariance predictions for Lanczos coefficients and their impact on long-time dynamics (survival probabilities, spectral form factor) and spread complexity, the authors connect Krylov-space growth to RMT universality. They show that for initial states with continuous energy support, plateau values depend only on the density of states, while random initial states reveal chaos-versus-integrability distinctions via eigenstate complexity and Anderson localization effects. Extending to generalized Dyson index ensembles, they reveal how universality classes shape the slope-to-plateau transition in spread complexity, suggesting deep links between Krylov dynamics, chaos, and quantum gravity questions.

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.

Figures (10)

• Figure 1: Variance of the Lanczos coefficients for random initial states for a Hamiltonian with Poisson distributed eigenvalues (solid blue for variance in $a_n$, orange for $4$ times variance in $b_n$), and for Hamiltonians chosen with Wigner-surmise-distributed \ref{['wign_surm']} spacing (green for variance in $a_n$, purple for $4$ times variance in $b_n$), compared to analytical predictions of the corresponding values for random matrices with the same density of states (light blue), with Dyson index $\beta_D=1$. These plots show that the Wigner surmise alone is not sufficient to produce the Lanczos spectrum of a RMT.
• Figure 2: The analytical predictions (via 8th order Cheybyshev approximations of the density of states) of the mean of the Lanczos spectrum (blue for $a_n$, orange for $b_n$) from the density of states, compared to the actual values of the Lanczos spectrum (black), for the four Hamiltonians in \ref{['spin_chain_ham']}. The matching is perfect for all models (integrable and chaotic).
• Figure 3: The analytical predictions (via 8th order Cheybyshev approximations of the density of states) of the variance of the Lanczos coefficients (black) compared to the numerically estimated variance of the Lanczos coefficients (blue for $a_n$, orange for $4$ times the variance of $b_n$), for the four Hamiltonians in \ref{['spin_chain_ham']}. We see that realistic chaotic systems conform with random matrix theory predictions of the variance of the Lanczos spectrum while integrable systems do not.
• Figure 4: Survival amplitudes, averaged over $256$ instances of a random initial state evolved with an $N=1024$ random matrix drawn from a distribution with the labeled potentials with Dyson index $2$. The orange graphs are the values obtained from a stretched spectrum while the blue graphs are the values from tridiagonal matrices following the analytical mean and covariance. Top row is without the edge correction and bottom row is with the edge correction, changing the first $n=7$ Lanczos coefficients. This demonstrates the necessity and accuracy of the edge correction to our analytical formula.
• Figure 5: The spread complexity of a random initial state for an $N=1024$ random matrix with the potential in \ref{['pot']}, averaged over 256 samples. Results from a stretched spectrum (orange) and using the analytical mean and covariance of the Lanzcos coefficients (blue). Top is without the edge correction, bottom is with the edge correction up to $n=7$. This demonstrates the necessity and accuracy of the edge correction to our analytical formula.