Abstract There is an abundance of evidence that some relaxation dynamics, e.g., exponential decays, are much more common in nature than others. Recently, there have been attempts to trace this dominance back to a certain stability of the prevalent dynamics versus generic Hamiltonian perturbations. In the paper at hand, we tackle this stability issue from yet another angle, namely in the framework of the recursion method. We investigate the behavior of various relaxation dynamics with respect to alterations of the so-called Lanczos coefficients. All considered scenarios are set up in order to comply with the ‘universal operator growth hypothesis’. Our numerical experiments suggest the existence of stability in a larger class of relaxation dynamics consisting of exponentially damped oscillations. Further, we propose a criterion to identify ‘pathological’ perturbations that lead to uncommon dynamics.
Abstract There is an abundance of evidence that some relaxation dynamics, e.g., exponential decays, are much more common in nature than others. Recently, there have been attempts to trace this dominance back to a certain stability of the prevalent dynamics versus generic Hamiltonian perturbations. In the paper at hand, we tackle this stability issue from yet another angle, namely in the framework of the recursion method. We investigate the behavior of various relaxation dynamics with respect to alterations of the so-called Lanczos coefficients. All considered scenarios are set up in order to comply with the ‘universal operator growth hypothesis’. Our numerical experiments suggest the existence of stability in a larger class of relaxation dynamics consisting of exponentially damped oscillations. Further, we propose a criterion to identify ‘pathological’ perturbations that lead to uncommon dynamics.
Concepts like ``typicality'' and the ``eigenstate thermalization hypothesis'' aim at explaining the apparent equilibration of quantum systems, possibly after a very long time. However, these concepts are not concerned with the specific way in which this equilibrium is approached. Our point of departure is the (evident) observation that some forms of the approach to equilibrium, such as, e.g., exponential decay of observables, are much more common than others. We suggest to trace this dominance of certain decay dynamics back to a larger stability with respect to generic Hamiltonian perturbations. A numerical study of a number of examples in which both the unperturbed Hamiltonians and the perturbations are modeled by partially random matrices is presented. We furthermore develop a simple heuristic, mathematical scheme that describes the result of the numerical investigations remarkably well. According to those investigations, the exponential decay indeed appears to be most stable. Dynamics that are in a certain sense at odds with the arrow of time are found to be very unstable.
Abstract Evidently, some relaxation dynamics, e.g. exponential decays, are much more common in nature than others. Recently there have been attempts to explain this observation on the basis of “typicality of perturbations” with respect to their impact on expectation value dynamics. These theories suggest that a majority of the very numerous, possible Hamiltonian perturbations entail more or less the same type of alteration of the decay dynamics. Thus, in this paper, we study how the approach towards equilibrium in closed quantum systems is altered due to weak perturbations. To this end, we perform numerical experiments on a particular, exemplary spin system. We compare our numerical data to predictions from three particular theories. We find satisfying agreement in the weak perturbation regime for one of these approaches.
We present a hypothesis for the universal properties of operators evolving under Hamiltonian dynamics in many-body systems.The hypothesis states that successive Lanczos coefficients in the continued fraction expansion of the Green's functions grow linearly with rate α in generic systems, with an extra logarithmic correction in 1D.The rate α-an experimental observable-governs the exponential growth of operator complexity in a sense we make precise.This exponential growth prevails beyond semiclassical or large-N limits.Moreover, α upper bounds a large class of operator complexity measures, including the out-of-timeorder correlator.As a result, we obtain a sharp bound on Lyapunov exponents λ L ≤ 2α, which complements and improves the known universal low-temperature bound λ L ≤ 2πT.We illustrate our results in paradigmatic examples such as nonintegrable spin chains, the Sachdev-Ye-Kitaev model, and classical models.Finally, we use the hypothesis in conjunction with the recursion method to develop a technique for computing diffusion constants.
A hypothesis is presented for the universal properties of operators evolving under Hamiltonian dynamics in many-body systems. The hypothesis states that successive Lanczos coefficients in the continued fraction expansion of the Green's functions grow linearly with rate alpha in generic systems, with an extra logarithmic correction in 1d. The rate alpha --- an experimental observable --- governs the exponential growth of operator complexity in a sense that is made precise. This exponential growth prevails beyond semiclassical or large-N limits. Moreover, alpha upper bounds a large class of operator complexity measures, including the out-of-time-order correlator. As a result, a sharp bound is obtained on Lyapunov exponents lambdaLleq2alpha, which complements and improves the known universal low-temperature bound lambdaLleq2piT. Our results are illustrated in paradigmatic examples such as non-integrable spin chains, the Sachdev-Ye-Kitaev model, and classical models. Finally hypothesis is used in in conjunction with the recursion method to develop a technique for computing diffusion constants.
Recently, a hypothesis on the complexity growth of unitarily evolving operators was presented. This hypothesis states that in generic, nonintegrable many-body systems, the so-called Lanczos coefficients associated with an autocorrelation function grow asymptotically linear, with a logarithmic correction in one-dimensional systems. In contrast, the growth is expected to be slower in integrable or free models. In this paper, we numerically test this hypothesis for a variety of exemplary systems, including one-dimensional and two-dimensional Ising models as well as one-dimensional Heisenberg models. While we find the hypothesis to be practically fulfilled for all considered Ising models, the onset of the hypothesized universal behavior could not be observed in the attainable numerical data for the Heisenberg model. The proposed linear bound on operator growth associated with the hypothesis eventually stems from geometric arguments involving the locality of the Hamiltonian as well as the lattice configuration. We derive and investigate a related geometric bound, and we find that while the bound itself is not sharply achieved for any considered model, the hypothesis is nonetheless fulfilled in most cases.
This cumulative doctoral thesis, based on the works [P1–P5], addresses various topical concepts as well as foundational questions in the nonequilibrium physics of quantum many-body systems. Publication [P1] concerns itself with the question of time’s arrow in quantum mechanics. Based on the recursion method, we attribute autocorrelation functions of few-body observables at infinite temperature (almost) monotonously decreasing functions, constructed from temporal derivatives of the correlation function that indicate a directed approach to equilibrium. We numerically find that in chaotic models, these constructions comprise a notion of simplicity of the underlying dynamics. Addressing the ubiquity of simple relaxation dynamics in otherwise complex chaotic systems, in Ref. [P3] we introduce a scheme based on Lanczos coefficients to approximate correlation functions in the thermodynamic limit. Examining several pertinent condensed-matter type models and benchmarking with state-of-the-art numerics, we find the method to yield good results already for a low number of included Lanczos coefficients, provided that the latter exhibit a smooth and increasing structure. We underpin these findings with analytical arguments on the method’s applicability. Ref. [P2] provides an estimate on the equilibration time of such infinite-temperature correlation functions in the thermodynamic limit. Similarly to Ref. [P3], we find the estimate to capture the equilibration times very reasonably, given that the corresponding Lanczos coefficients exhibit a smooth growth. Substantiated by analytical considerations, the convergence of the estimate implies that equilibration timescales of the relaxation processes are finite and conceivably short. In Ref. [P5] we scrutinize the interpretative power of the Lanczos coefficients concerning the system’s integrability or chaoticity, operator growth and the indicativeness of physical behavior. To this end, we analytically deter- mine the Lanczos coefficients for a Majorana observable in a strictly quadratic model and illustrate that within this model any structure of the Lanczos coefficients may be realised. Different to the other works, Ref. [P4] is not cast in the language of the recursion method. Focusing on the full eigenstate thermalization hypothesis, it explores the connection between thermalization dynamics of multi-point correlation functions and the scaling of thermal free cumulants in setups mimicking a system-bath situation.