Estimation of equilibration time scales from nested fraction approximations

Abstract

We consider an autocorrelation function of a quantum mechanical system through the lens of the so-called recursive method, by iteratively evaluating Lanczos coefficients, or solving a system of coupled differential equations in the Mori formalism. We first show that both methods are mathematically equivalent, each offering certain practical advantages. We then propose an approximation scheme to evaluate the autocorrelation function, and use it to estimate the equilibration time $\tau$ for the observable in question. With only a handful of Lanczos coefficients as the input, this scheme yields an accurate order of magnitude estimate of $\tau$, matching state-of-the-art numerical approaches. We develop a simple numerical scheme to estimate the precision of our method. We test our approach using several numerical examples exhibiting different relaxation dynamics. Our findings provide a new practical way to quantify the equilibration time of isolated quantum systems, a question which is both crucial and notoriously difficult.

arXiv comment9 pages, 11 figures

References in corpus (21)

• Comments on the Sachdev-Ye-Kitaev model
• Equilibration, thermalisation, and the emergence of statistical mechanics in closed quantum systems
• Foundation of Statistical Mechanics under experimentally realistic conditions
• A Universal Operator Growth Hypothesis
• Toda chain flow in Krylov space
• Typical fast thermalization processes in closed many-body systems
• On the time scales in the approach to equilibrium of macroscopic quantum systems
• Quantum Systems Equilibrate Rapidly for Most Observables
• Equilibration time scales of physically relevant observables
• Convergence to equilibrium under a random Hamiltonian
• Extremely quick thermalization in a macroscopic quantum system for a typical nonequilibrium subspace
• The complexity of energy eigenstates as a mechanism for equilibration
• Eigenstate thermalization hypothesis and its deviations from random-matrix theory beyond the thermalization time
• Krylov complexity in quantum field theory, and beyond
• Time evolution of correlation functions in quantum many-body systems
• Subsystem dynamics under random Hamiltonian evolution
• Selected applications of typicality to real-time dynamics of quantum many-body systems
• Impact of eigenstate thermalization on the route to equilibrium
• Universal Hypothesis of Autocorrelation Function from Krylov Complexity
• Modern Concepts of Quantum Equilibration Do Not Rule Out Strange Relaxation Dynamics
• Relaxation of dynamically prepared out-of-equilibrium initial states within and beyond linear response theory

Cited by in corpus (8)

• Quantum Dynamics in Krylov Space: Methods and Applications
• Pseudomode expansion of many-body correlation functions
• Lanczos-Pascal approach to correlation functions in chaotic quantum systems
• Estimate of equilibration times of quantum correlation functions in the thermodynamic limit based on Lanczos coefficients
• Scrutinizing the Mori memory function for diffusion in periodic quantum systems
• Hierarchy of timescales in a disordered spin-$1/2$ XX ladder
• Nontrivial damping of magnetization currents in perturbed spin chains
• Resonant level model from a Krylov perspective: Lanczos coefficients in a quadratic model