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arXiv:2305.02356 · Quantum Frontiers

Universal hypothesis of autocorrelation function from Krylov complexity

Ren Zhang, Hui Zhai

Published Apr 11, 2024• 14 citations
arXivHTMLPDF

Abstract

Abstract In a quantum many-body system, autocorrelation functions can determine linear responses nearby equilibrium and quantum dynamics far from equilibrium. In this letter, we bring out the connection between the operator complexity and the autocorrelation function. In particular, we focus on a particular kind of operator complexity called the Krylov complexity. We find that a set of Lanczos coefficients $\{b_{n}\}$ { b n } computed for determining the Krylov complexity can reveal the universal behaviors of autocorrelations, which are otherwise impossible. When the time axis is scaled by $b_{1}$ b 1 , different autocorrelation functions obey a universal function form at short time. We further propose a characteristic parameter deduced from $\{b_{n}\}$ { b n } that can largely determine the behavior of autocorrelations at the intermediate time. This parameter can also largely determine whether the autocorrelation function oscillates or monotonically decays in time. We present numerical evidences and physical intuitions to support these universal hypotheses of autocorrelations. We emphasize that these universal behaviors are held across different operators and different physical systems.

DOI
10.1007/s44214-024-00054-4

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Abstract In a quantum many-body system, autocorrelation functions can determine linear responses nearby equilibrium and quantum dynamics far from equilibrium. In this letter, we bring out the connection between the operator complexity and the autocorrelation function. In particular, we focus on a particular kind of operator complexity called the Krylov complexity. We find that a set of Lanczos coefficients {bn}\{b_{n}\}{bn​} { b n } computed for determining the Krylov complexity can reveal the universal behaviors of autocorrelations, which are otherwise impossible. When the time axis is scaled by b1b_{1}b1​ b 1 , different autocorrelation functions obey a universal function form at short time. We further propose a characteristic parameter deduced from {bn}\{b_{n}\}{bn​} { b n } that can largely determine the behavior of autocorrelations at the intermediate time. This parameter can also largely determine whether the autocorrelation function oscillates or monotonically decays in time. We present numerical evidences and physical intuitions to support these universal hypotheses of autocorrelations. We emphasize that these universal behaviors are held across different operators and different physical systems.
A bstract We investigate the Krylov complexity of Schrödinger field theories, focusing on both bosonic and fermionic systems within the grand canonical ensemble which includes a chemical potential. Krylov complexity measures operator growth in quantum systems by analyzing how operators spread within the Krylov space, a subspace of the Hilbert space spanned by successive applications of the superoperator [ H , ·] on an initial operator. Using the Lanczos algorithm, we construct an orthonormal Krylov basis and derive the Lanczos coefficients, which govern the operator connectivity and thus characterize the complexity. Our study reveals that the Lanczos coefficients { b n } are almost independent of the chemical potential, while { a n } are dependent on the chemical potential. Both { a n } and { b n } show linear relationships with respect to n . For both bosonic and fermionic systems, the Krylov complexities behave similarly over time, especially at late times, due to the analogous profiles of the squared absolute values of their autocorrelation functions | φ 0 ( t )| 2 . The Krylov complexity grows exponentially with time, but its asymptotic scaling factor λ K is significantly smaller than the twice of the slope of the { b n } coefficients, contrasting to the relativistic field theories where the scaling aligns more closely with the twice of the slope of { b n }.
We study the universal properties of the Lanczos algorithm applied to finite-size many-body quantum systems. Focusing on autocorrelation functions of local operators and on their infinite-time behaviour at finite size, we conjecture that in the large nnn limit, the ratios between consecutive Lanczos coefficients should have specific scalings with the size of the lattice that we make precise and that depend on the hydrodynamic tail of the autocorrelation function. The scaling associated with strong or approximate zero-modes is also discussed. We support our conjecture with a numerical study of different models.
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.
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.
In the study of quantum chaos diagnostics, considerable attention has been attributed to the Krylov complexity and spectrum form factor (SFF) for systems at infinite temperature. These investigations have unveiled universal properties of quantum chaotic systems. By extending the analysis to include the finite temperature effects on the Krylov complexity and SFF, we demonstrate that the Lanczos coefficients bnb_nbn​, which are associated with the Wightman inner product, display consistency with the universal hypothesis presented in PRX 9, 041017 (2019). This result contrasts with the behavior of Lanczos coefficients associated with the standard inner product. Our results indicate that the slope ααα of the bnb_nbn​ is bounded by πkBTπk_BTπkB​T, where kBk_BkB​ is the Boltzmann constant and TTT the temperature. We also investigate the SFF, which characterizes the two-point correlation of the spectrum and encapsulates an indicator of ergodicity denoted by ggg in chaotic systems. Our analysis demonstrates that as the temperature decreases, the value of ggg decreases as well. Considering that ααα also represents the operator growth rate, we establish a quantitative relationship between ergodicity indicator and Lanczos coefficients slope. To support our findings, we provide evidence using the Gaussian orthogonal ensemble and a random spin model. Our work deepens the understanding of the finite temperature effects on Krylov complexity, SFF, and the connection between ergodicity and operator growth.
Local counterdiabatic (CD) driving provides a systematic way of constructing a control protocol to approximately suppress the excitations resulting from changing some parameter(s) of a quantum system at a finite rate. However, designing CD protocols typically requires knowledge of the original Hamiltonian . In this work, we design local CD driving protocols in Krylov space using only the characteristic local time scales of the system set by e.g., phonon frequencies in materials or Rabi frequencies in superconducting qubit arrays. Surprisingly, we find that convergence of these universal protocols is controlled by the asymptotic high-frequency tails of the response functions. This finding hints at a deep connection between the long-time, low-frequency response of the system controlling non-adiabatic effects, and the high-frequency response determined by the short-time operator growth and the Krylov complexity. We make this connection concrete by showing how, for a representative integrable model, we may extract long-time universal behavior of the correlation functions from a short-time expansion of the dynamics using a system-independent universal protocol.