Heating Rates in Periodically Driven Strongly Interacting Quantum Many-Body Systems

Abstract

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 regime in which heating is exponential in time. 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 frequency of the drive.

Entanglement and matrix elements of observables in interacting integrable systems

“…9(a) and 9(c), the same is equally true for the integrable case as conjectured in Ref. [43]. Armed with this knowledge, we can now extract the smooth function |f O (Ē 0, ω)| 2 , which is independent of system size for nonvanishing values of ω [1], that char- acterizes |O αβ | 2 .…”

“…We also studied the smooth function |f O (Ē 0, ω)| 2 that characterizes the variance and contrasted its behavior at interacting integrable and nonintegrable points. It was recently argued that this function can be measured in experiments with periodically driven systems, both nonintegrable and interacting integrable ones, by studying how heating rates change when changing the frequency of the drive [43].…”

“…A second point to be highlighted from the behavior of Γ O (ω) at integrability is that, since Γ O (ω) increases with increasing system size L, |O αβ | 2 decreases more slowly with increasing L than |O αβ | 2 . Since |O αβ | 2 is the quantity that enters in fluctuation dissipation relations [1,26], transport properties [27,83], and heating rates under periodic driving [43], in what follows we focus on the scaling of |O αβ | 2 with increasing system size, and on the smooth function that characterizes the dependence of |O αβ | 2 on ω.…”

“…For noninteracting models (or models mappable to them), the existence of an increasingly large (with increasing system size) fraction of vanishing off-diagonal matrix elements precludes the definition of a meaningful function f O (Ē, ω), in contrast to quantum chaotic models [26]. On the other hand, recent results by two of us (KM and MR) in the context of periodically driven systems provided strong evidence that one can define (and experimentally measure) a function |f O (Ē, ω)| 2 = e S(Ē) |O αβ | 2 for interacting integrable models [43]. Ex-arXiv:1909.09654v1 [cond-mat.stat-mech] 20 Sep 2019 ploring this, along with other properties of the offdiagonal (and diagonal) matrix elements in the spin-1/2 XXZ chain, is one of the two central goals of this work.…”

Eigenstate thermalization in dual-unitary quantum circuits: Asymptotics of spectral functions

“…While the small frequency behavior ω → 0 of the spectral function and the corresponding statistics of diagonal matrix elements A nn encodes equilibrium properties at large times t → ∞, finite frequencies |ω| > 0 and the statistics of off-diagonal elements determine fluctuations in equilibrium as well as the dynamics of relaxation to equilibrium. Consequently, the spectral functions governs the decay of dynamical correlations via fluctuationdissipation relations and linear response theory [49][50][51][52][53][54] yielding, e.g., heating rates in driven systems [55] and sensitive probes to quantum chaos [56,57].…”