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.
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.
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.
Out-of-time-ordered correlation functions (OTOCs) are presently being extensively debated as quantifiers of dynamical chaos in interacting quantum many-body systems. We argue that in quantum spin and fermionic systems, where all local operators are bounded, an OTOC of local observables is bounded as well and thus its exponential growth is merely transient. As a better measure of quantum chaos in such systems, we propose, and study, the density of the OTOC of extensive sums of local observables, which can exhibit indefinite growth in the thermodynamic limit. We demonstrate this for the kicked quantum Ising model by using large-scale numerical results and an analytic solution in the integrable regime. In a generic case, we observe the growth of the OTOC density to be linear in time. We prove that this density in general, locally interacting, nonintegrable quantum spin and fermionic dynamical systems exhibits growth that is at most polynomial in time---a phenomenon, which we term weak quantum chaos. In the special case of the model being integrable and the observables under consideration quadratic, the OTOC density saturates to a plateau.
Local out-of-time-order correlators (OTOCs) measure the growth of initially small commutators. For bounded local operators, this growth ends when the local commutator becomes of order one, but in an extended system the region over which it is large can continue to expand. We study this spatial proliferation using three related observables: the local squared commutator Cr(t), its spatial sum Cˉ(t)=∑rCr(t), and the symmetrically regularised connected dOTOC D(t) of two extensive observables. We show that Lieb--Robinson locality gives Cˉ(t)≲td. We then prove that, in a Gibbs state with uniform exponential clustering, the full dOTOC obeys the same bound, D(t)≲td, improving our earlier suboptimal t3d bound. We illustrate these results in two quantum many-body systems: a Brownian coupled-cluster model and a finite-temperature SYK chain. In both cases, an exact disorder symmetry relates D(t) to Cˉ(t) and allows us to follow the same extensive observable from the pre-scrambling Lyapunov regime into the post-scrambling growth associated with an expanding saturated region. This late-time power-law growth is the concrete realisation of the weak quantum chaos introduced in our earlier work.
Scrambling of information in a quantum many-body system, quantified by the out-of-time-ordered correlator (OTOC), is a key manifestation of quantum chaos. A regime of exponential growth in the OTOC, characterized by a Lyapunov exponent, has so far mostly been observed in systems with a high-dimensional local Hilbert space and in weakly-coupled systems. Here, we propose a general criterion for the existence of a well-defined regime of exponential growth of the OTOC in spatially extended systems with local interactions. In such systems, we show that a parametrically long period of exponential growth requires the butterfly velocity to be much larger than the Lyapunov exponent times a microscopic length scale, such as the lattice spacing. As an explicit example, we study a random unitary circuit with tunable interactions. In this model, we show that in the weakly interacting limit, the above criterion is satisfied, and there is a prolonged window of exponential growth. Our results are based on numerical simulations of both Clifford and universal random circuits supported by an analytical treatment.
A bstract In semi-classical systems, the exponential growth of the out-of-time-order correlator (OTOC) is believed to be the hallmark of quantum chaos. However, on several occasions, it has been argued that, even in integrable systems, OTOC can grow exponentially due to the presence of unstable saddle points in the phase space. In this work, we probe such an integrable system exhibiting saddle-dominated scrambling through Krylov complexity and the associated Lanczos coefficients. In the realm of the universal operator growth hypothesis, we demonstrate that the Lanczos coefficients follow the linear growth, which ensures the exponential behavior of Krylov complexity at early times. The linear growth arises entirely due to the saddle, which dominates other phase-space points even away from itself. Our results reveal that the exponential growth of Krylov complexity can be observed in integrable systems with saddle-dominated scrambling and thus need not be associated with the presence of chaos.
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.