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Local Operators and Quantum Chaos

Daniel E. Parker

Published Jan 1, 2020• 2 citations
OpenAlex

Abstract

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 $\\lambda_L \\leq 2 \\alpha$, which complements and improves the known universal low-temperature bound $\\lambda_L \\leq 2 \\pi T$. 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.

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Krylov complexity, as a novel measure of operator complexity under Heisenberg evolution, exhibits many interesting universal behaviors and also bounds many other complexity measures. In this work, we study Krylov complexity $\mathcal{K}(t)$ in Random Matrix Theory (RMT). In large $N$ limit: (1) For infinite temperature, we analytically show that the Lanczos coefficient $\{b_n\}$ saturate to constant plateau $\lim\limits_{n\rightarrow\infty}b_n=b$, rendering a linear growing complexity $\mathcal{K}(t)\sim t$, in contrast to the exponential-in-time growth in chaotic local systems in thermodynamic limit. After numerically comparing this plateau value $b$ to a large class of chaotic local quantum systems, we find that up to small fluctuations, it actually bounds the $\{b_n\}$ in chaotic local quantum systems. Therefore we conjecture that in chaotic local quantum systems after scrambling time, the speed of linear growth of Krylov complexity cannot be larger than that in RMT. (2) For low temperature, we analytically show that $b_n$ will first exhibit linear growth with $n$, whose slope saturates the famous chaos bound. After hitting the same plateau $b$, $b_n$ will then remain constant. This indicates $\mathcal{K}(t)\sim e^{2πt/β}$ before scrambling time $t_*\sim O(β\logβ)$, and after that it will grow linearly in time, with the same speed as in infinite temperature. We finally remark on the effect of finite $N$ corrections.

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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\\alphaalpha in generic systems, with an extra logarithmic correction in 1d. The rate alpha\\alphaalpha --- 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-NNN limits. Moreover, alpha\\alphaalpha 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\\lambda_L \\leq 2 \\alphalambdaL​leq2alpha, which complements and improves the known universal low-temperature bound lambdaLleq2piT\\lambda_L \\leq 2 \\pi TlambdaL​leq2piT. 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.
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.
We analyze local operator growth in nonintegrable quantum many-body systems. A recently introduced universal operator growth hypothesis proposes that the maximal growth of Lanczos coefficients in the continued fraction expansion of the Green's function reflects chaos of the underlying system. We first show that the continued fraction expansion, and the recursion method in general, should be understood in the context of a completely integrable classical dynamics in Krylov space. In particular, the time-correlation function of a physical observable analytically continued to imaginary time is a tau-function of integrable Toda hierarchy. We use this relation to generalize the universal operator growth hypothesis to include arbitrarily ordered correlation functions. We then proceed to analyze the singularity of the time-correlation function, which is an equivalent sign of chaos to the maximal growth of Lanczos coefficients, and we show that it is due to delocalization in Krylov space. We illustrate the general relation between chaos and delocalization using an explicit example of the Sachdev-Ye-Kietaev model.
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.
We consider growth of local operators under Euclidean time evolution in lattice systems with local interactions.We derive rigorous bounds on the operator norm growth and then proceed to establish an analog of the Lieb-Robinson bound for the spatial growth.In contrast to the Minkowski case when ballistic spreading of operators is universal, in the Euclidean case spatial growth is system-dependent and indicates if the system is integrable or chaotic.In the integrable case, the Euclidean spatial growth is at most polynomial.In the chaotic case, it is the fastest possible: exponential in 1D, while in higher dimensions and on Bethe lattices local operators can reach spatial infinity in finite Euclidean time.We use bounds on the Euclidean growth to establish constraints on individual matrix elements and operator power spectrum.We show that one-dimensional systems are special with the power spectrum always being superexponentially suppressed at large frequencies.Finally, we relate the bound on the Euclidean growth to the bound on the growth of Lanczos coefficients.To that end, we develop a path integral formalism for the weighted Dyck paths and evaluate it using saddle point approximation.Using a conjectural connection between the growth of the Lanczos coefficients and the Lyapunov exponent controlling the growth of the out-of-time-ordered correlators (OTOCs), we propose an improved bound on chaos valid at all temperatures.
The operator growth hypothesis (OGH) is a technical conjecture about the behavior of operators—specifically, the asymptotic growth of their —under repeated action by a Liouvillian. It is expected to hold for a sufficiently generic closed many-body system. When it holds, it yields bounds on the high-frequency behavior of local correlation functions and measures of chaos (like OTOCs). It also gives a route to numerically estimating response functions. Here we investigate the generalization of OGH to open quantum systems, where the Liouvillian is replaced by a Lindbladian. For a quantum system with local Hermitian jump operators, we show that the OGH is modified: we define a generalization of the Lanczos coefficient and show that it initially grows linearly as in the original OGH, but experiences exponentially growing oscillations on scales determined by the dissipation strength. We see this behavior manifested in a semi-analytically solvable model (large- q SYK with dissipation), numerically for an ergodic spin chain, and in a solvable toy model for operator growth in the presence of dissipation (which resembles a non-Hermitian single-particle hopping process). Finally, we show that the modified OGH connects to a fundamental difference between Lindblad and closed systems: At high frequencies, the spectral functions of the former decay algebraically, while in the latter they decay exponentially. This is an experimentally testable statement, which also places limitations on the applicability of Lindbladians to systems in contact with equilibrium environments. Published by the American Physical Society 2024
Krylov complexity, as a novel measure of operator complexity under Heisenberg evolution, exhibits many interesting universal behaviors and also bounds many other complexity measures. In this work, we study Krylov complexity K(t)\mathcal{K}(t)K(t) in Random Matrix Theory (RMT). In large NNN limit: (1) For infinite temperature, we analytically show that the Lanczos coefficient {bn}\{b_n\}{bn​} saturate to constant plateau lim⁡n→∞bn=b\lim\limits_{n\rightarrow\infty}b_n=bn→∞lim​bn​=b, rendering a linear growing complexity K(t)∼t\mathcal{K}(t)\sim tK(t)∼t, in contrast to the exponential-in-time growth in chaotic local systems in thermodynamic limit. After numerically comparing this plateau value bbb to a large class of chaotic local quantum systems, we find that up to small fluctuations, it actually bounds the {bn}\{b_n\}{bn​} in chaotic local quantum systems. Therefore we conjecture that in chaotic local quantum systems after scrambling time, the speed of linear growth of Krylov complexity cannot be larger than that in RMT. (2) For low temperature, we analytically show that bnb_nbn​ will first exhibit linear growth with nnn, whose slope saturates the famous chaos bound. After hitting the same plateau bbb, bnb_nbn​ will then remain constant. This indicates K(t)∼e2πt/β\mathcal{K}(t)\sim e^{2πt/β}K(t)∼e2πt/β before scrambling time t∗∼O(βlog⁡β)t_*\sim O(β\logβ)t∗​∼O(βlogβ), and after that it will grow linearly in time, with the same speed as in infinite temperature. We finally remark on the effect of finite NNN corrections.