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 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.
Abstract The universal operator-growth hypothesis (OGH) states that in a closed chaotic system the Lanc- zos coeffcients grow linearly, bn ≃ αn, and that the rate α bounds the Lyapunov exponent, λL ≤ 2α. We ask how this structure is modified when the system is coupled to a Markovian environment, so that the generator becomes non-Hermitian. Applying the Arnoldi recursion to the vectorized Lindbladian in the infinite-temperature Wightman inner product, we organize the resulting com- plex growth rate αC = αR + iαI (the real sub-diagonal and complex diagonal Arnoldi coeffcients) around three exact statements. First, the diagonal’s real part is fixed exactly by Re an = −2γ⟨ ˆ G⟩n, identifying the imaginary rate as the rate at which the growing operator accumulates weight in the dissipation channels; its magnitude σG is non-universal (set by the jump operators), but the generator symmetry G† (γ) = −G(−γ) makes bn even and Re an odd in the dissipation strength γ, so universally αI = 2γσG turns on linearly and αR is renormalized only at O(γ2 ). Second, for the large-q SYK model we obtain the closed form αC = J − 2i(q − 2)γ, whose imaginary part is fixed solely by the interaction range. Third, we argue—on the same footing as the OGH itself—that the bound extends as λL ≤ 2αR(γ) with αR(γ) ≤ αR(0) + O(γ), because the dissipator is a local bounded superoperator that narrows the autocorrelation’s analyticity strip by at most O(γ). Rather than merely acquiring a phase, the imaginary rate is an exact, parity-odd measure of dissipative operator spreading, while the complexity-growth rate is protected to O(γ2 ): dissipation does not suppress operator growth.
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.
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
A bstract Quantum observables of generic many-body systems exhibit a universal pattern of growth in the Krylov space of operators. This pattern becomes particularly manifest in the Lanczos basis, where the evolution superoperator assumes the tridiagonal form. According to the universal operator growth hypothesis, the nonzero elements of the superoperator, known as Lanczos coefficients, grow asymptotically linearly. We introduce and explore broad families of Lanczos coefficients that are consistent with the universal operator growth and lead to the exactly solvable dynamics. Within these families, the subleading terms of asymptotic expansion of the Lanczos sequence can be controlled and fine-tuned to produce diverse dynamical patterns. For one of the families, the Krylov complexity is computed exactly.