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arXiv:2109.01705 · Physical review. B./Physical review. B

Prethermalization, thermalization, and Fermi's golden rule in quantum many-body systems

Krishnanand Mallayya, Marcos Rigol

Published Nov 15, 2021• 22 citations
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Abstract

We study the prethermalization and thermalization dynamics of local observables in weakly perturbed nonintegrable systems, with Hamiltonians of the form ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{0}+g\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{V}$, where ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{0}$ is nonintegrable and $g\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{V}$ is a perturbation. We explore the dynamics of far from equilibrium initial states in the thermodynamic limit using a numerical linked cluster expansion (NLCE), and in finite systems with periodic boundaries using exact diagonalization. We argue that generic observables exhibit a two-step relaxation process, with a fast prethermal dynamics followed by a slow thermalizing one, only if the perturbation breaks a conserved quantity of ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{0}$ and if the value of the conserved quantity in the initial state is $\mathcal{O}(1)$ different from the one after thermalization. We show that the slow thermalizing dynamics is characterized by a rate $\ensuremath{\propto}{g}^{2}$, which can be accurately determined using a Fermi golden rule (FGR) equation. We also show that during such a slow dynamics, observables can be described using projected diagonal and Gibbs ensembles, and we contrast their accuracy.

DOI
10.1103/physrevb.104.184302

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We study the prethermalization and thermalization dynamics of local observables in weakly perturbed nonintegrable systems, with Hamiltonians of the form H\ifmmode^\elseˆ\fi0+gV\ifmmode^\elseˆ\fi{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{0}+g\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{V}H\ifmmode^\elseˆ\fi​0​+gV\ifmmode^\elseˆ\fi​, where H\ifmmode^\elseˆ\fi0{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{0}H\ifmmode^\elseˆ\fi​0​ is nonintegrable and gV\ifmmode^\elseˆ\fig\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{V}gV\ifmmode^\elseˆ\fi​ is a perturbation. We explore the dynamics of far from equilibrium initial states in the thermodynamic limit using a numerical linked cluster expansion (NLCE), and in finite systems with periodic boundaries using exact diagonalization. We argue that generic observables exhibit a two-step relaxation process, with a fast prethermal dynamics followed by a slow thermalizing one, only if the perturbation breaks a conserved quantity of H\ifmmode^\elseˆ\fi0{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{0}H\ifmmode^\elseˆ\fi​0​ and if the value of the conserved quantity in the initial state is O(1)\mathcal{O}(1)O(1) different from the one after thermalization. We show that the slow thermalizing dynamics is characterized by a rate \ensuremath∝g2\ensuremath{\propto}{g}^{2}\ensuremath∝g2, which can be accurately determined using a Fermi golden rule (FGR) equation. We also show that during such a slow dynamics, observables can be described using projected diagonal and Gibbs ensembles, and we contrast their accuracy.
We investigate the crossover of the entanglement entropy toward its thermal value in nearly integrable systems. We employ equations-of-motion techniques to study the entanglement dynamics in a lattice model of weakly interacting spinless fermions after a quantum quench. For weak enough interactions we observe a two-step relaxation of the entanglement entropies of finite subsystems. Initially, the entropies follow a nearly integrable evolution, approaching the value predicted by the generalized Gibbs ensemble (GGE) of the unperturbed model. Then, they start a slow drift toward the thermal stationary value described by a standard Gibbs ensemble (GE). While the initial relaxation to the GGE is independent of the interaction, the slow drift from GGE to GE values happens on timescales proportional to the inverse interaction squared. For asymptotically large times and subsystem sizes the dynamics of the entropies can be predicted using a modified quasiparticle picture that keeps track of the evolution of the fermionic occupations caused by the integrability breaking. This picture gives a quantitative description of the results as long as the integrability-breaking timescale is much larger than the one associated with the (quasi)saturation to the GGE. In the opposite limit, the quasiparticle picture still provides the correct late-time behavior, but it underestimates the initial slope of the entanglement entropy.
We study heating rates in strongly interacting quantum lattice systems in the thermodynamic limit. Using a numerical linked cluster expansion, we calculate the energy as a function of the driving time and find a robust exponential regime. The heating rates are shown to be in excellent agreement with Fermi's golden rule. We discuss the relationship between heating rates and, within the eigenstate thermalization hypothesis, the smooth function that characterizes the off-diagonal matrix elements of the drive operator in the eigenbasis of the static Hamiltonian. We show that such a function, in nonintegrable and (remarkably) integrable Hamiltonians, can be probed experimentally by studying heating rates as functions of the drive frequency.
Nonequilibrium dynamics of a nonintegrable system without the eigenstate thermalization hypothesis is studied. It is shown that, in the thermodynamic limit, this model thermalizes after an arbitrary quantum quench at finite temperature, although it does not satisfy the eigenstate thermalization hypothesis. In contrast, when the system size is finite and the temperature is low enough, the system may not thermalize. In this case, the steady state is well described by the generalized Gibbs ensemble constructed by using highly nonlocal conserved quantities. We also show that this model exhibits prethermalization, in which the prethermalized state is characterized by nonthermal energy eigenstates.
We study heating dynamics in isolated quantum many-body systems driven periodically at high frequency and large amplitude. Combining the high-frequency expansion for the Floquet Hamiltonian with Fermi's golden rule (FGR), we develop a master equation termed the Floquet FGR. Unlike the conventional one, the Floquet FGR correctly describes heating dynamics, including the prethermalization regime, even for strong drives, under which the Floquet Hamiltonian is significantly dressed, and nontrivial Floquet engineering is present. The Floquet FGR depends on system size only weakly, enabling us to analyze the thermodynamic limit with small-system calculations. Our results also indicate that, during heating, the system approximately stays in the thermal state for the Floquet Hamiltonian with a gradually rising temperature.
Although statistical mechanics describes thermal equilibrium states, these states may or may not emerge dynamically for a subsystem of an isolated quantum many-body system. For instance, quantum systems that are near-integrable usually fail to thermalize in an experimentally realistic time scale, and instead relax to quasi-stationary prethermal states that can be described by statistical mechanics, when approximately conserved quantities are included in a generalized Gibbs ensemble (GGE). We experimentally study the relaxation dynamics of a chain of up to 22 spins evolving under a long-range transverse-field Ising Hamiltonian following a sudden quench. For sufficiently long-range interactions, the system relaxes to a new type of prethermal state that retains a strong memory of the initial conditions. However, the prethermal state in this case cannot be described by a standard GGE; it rather arises from an emergent double-well potential felt by the spin excitations. This result shows that prethermalization occurs in a broader context than previously thought, and reveals new challenges for a generic understanding of the thermalization of quantum systems, particularly in the presence of long-range interactions.
We investigate a class of periodically driven many-body systems that allows us to extend the phenomenon of prethermalization to the vicinity of isolated intermediate-to-low drive frequencies away from the high-frequency limit. We provide numerical evidence for the formation of a parametrically long-lived prethermal plateau, captured by an effective Floquet Hamiltonian computed using the replica inverse-frequency expansion, and demonstrate its stability with respect to random perturbations in the drive period. Considering exclusively nonintegrable Floquet Hamiltonians, we find that heating rates are nonuniversal: we observe Fermi's golden rule scaling, power-law scaling inconsistent with the golden rule, and non-power-law scaling, depending on the drive. Despite the asymptotic character of the inverse-frequency expansion, we show that it describes the thermostatic properties of the state all along the evolution up to infinite temperature, with higher-order terms improving the accuracy. Our results suggest a dynamical mechanism to gradually increase the temperature in isolated quantum simulators, such as ultracold atoms, and open up an alternative possibility to investigate thermal phase transitions and the interplay between thermal and quantum criticality using Floquet drives.