Krylov space dynamics of ergodic and dynamically frozen Floquet systems

TL;DR

This work addresses the challenge of characterizing late-time dynamics and ergodicity breaking in periodically driven quantum many-body systems. It introduces a Krylov-space framework that projects the Floquet evolution onto a subspace generated by successive applications of the one-period evolution operator to the initial state, enabling efficient computation of infinite-time averages in the diagonal ensemble (DEA) via Ritz vectors. The authors demonstrate the method on a driven mixed-field Ising model, revealing a transition between ergodic and dynamically frozen phases and showing that Ritz-vector localization serves as a robust diagnostic of ergodicity; the approach also extends to Floquet-MBL scenarios. Overall, the Krylov DEA algorithm provides a scalable, accurate tool to study nonequilibrium steady states and ergodicity-breaking phenomena in Floquet matter, offering a compact reduced basis for analyzing long-time dynamics.

Abstract

In isolated quantum many-body systems periodically driven in time, the asymptotic dynamics at late times can exhibit distinct behavior such as thermalization or dynamical freezing. Understanding the properties of and the convergence towards infinite-time (nonequilibrium) steady states however remains a challenging endeavor. We propose a physically motivated Krylov space perspective on Floquet thermalization which offers a natural framework to study rates of convergence towards steady states and, simultaneously, an efficient numerical algorithm to evaluate infinite-time averages of observables within the diagonal ensemble. The effectiveness of our algorithm is demonstrated by applying it to the periodically driven mixed-field Ising model, reaching system sizes of up to 30 spins. Our method successfully resolves the transition between the ergodic and dynamically frozen phases and provides insight into the nature of the Floquet eigenstates across the phase diagram. Furthermore, we show that the long-time behavior is encoded within the localization properties of the Ritz vectors under the Floquet evolution, providing an accurate diagnostic of ergodicity.

Figures (6)

• Figure 1: We introduce a Krylov space approximation for computing the infinite-time average (“DEA”) of observables in periodically driven systems. The figure illustrates two contrasting scenarios: (left) an ergodic regime, where observables rapidly relax to equilibrium, and (right) a dynamically frozen regime, where emergent conservation laws keep observables locked near their initial values for long times. Our Krylov subspace approach yields accurate infinite-time averages in both cases.
• Figure 2: Infinite-time average of the total magnetization density, $m_z$, showing the transition to a dynamically frozen regime at large $H_d/J$. The solid line shows the exact long-time average for a 20-site chain. Crosses denote the same quantity computed with our algorithm after 150 iterations also for the same system size and circles mark converged results for a 30-site chain. Inset shows the error $\epsilon_{150}$ after 150 iterations of our Krylov algorithm with lighter to darker showing increasing system sizes. Dashed lines show a suggestive scaling of the error in the ergodic regime.
• Figure 3: The error $\epsilon_m$ of the DEA at each step during the first 150 iterations of the algorithm. Simulations were performed on a 20-site spin-chain, with red and blue lines indicating ergodic and dynamically frozen model parameters respectively. The dashed line shows a suggestive scaling for the error in the ergodic regime.
• Figure 4: Inverse participation ratios. (a) The inverse participation ratio of the initial state in the basis of Ritz vectors (approximate Floquet eigenvectors) for different Krylov space dimensions. Blue and red lines show frozen and ergodic model parameters respectively; the initial state can be seen to go between localized to delocalized in the Krylov subspace as the drive strength is increased. (b) The inverse participation ratio of the Ritz vectors in the computational basis, averaged over all Ritz vectors in a Krylov subspace of dimension 50. The dashed lines correspond to $1/m$ and $1/D$ in (a) and (b) respectively. (c) We fit the lines from (b) to the form $\sim D^{-\alpha}$ and show the scaling exponent, with error bars, for different drive strengths $H_d/J$. Different lines correspond to different Krylov subspace dimensions, $m$. A sharp transition between localized and delocalized Ritz vectors in the full Hilbert space can be seen. The inset shows a more fine-grained parameter scan nearer the transition (purple shaded region of the main plot).
• Figure 5: We compare different numerical approaches to computing the DEA with our Krylov subspace algorithm. Dashed line denotes the exact DEA for the magnetization density on a spin-chain with 12 sites and $H_d/J=0.5$. Generalized eigensolver (green line) denotes the algorithm choice used in the main text where we take the right eigenvectors of $U_m$. Purple line shows the DEA computed with the eigenvectors of $U_m + U_m^\dagger$ and the pink line shows the DEA computed using an isometric Arnoldi procedure.