Out-of-time ordered (OTO) correlation functions describe scrambling of information in correlated quantum matter. They are of particular interest in incoherent quantum systems lacking well defined quasi-particles. Thus far, it is largely elusive how OTO correlators spread in incoherent systems with diffusive transport governed by a few globally conserved quantities. Here, we study the dynamical response of such a system using high-performance matrix-product-operator techniques. Specifically, we consider the non-integrable, one-dimensional Bose-Hubbard model in the incoherent hightemperature regime. Our system exhibits diffusive dynamics in time-ordered correlators of globally conserved quantities, whereas OTO correlators display a ballistic, light-cone spreading of quantum information. The slowest process in the global thermalization of the system is thus diffusive, yet information spreading is not inhibited by such slow dynamics. We furthermore develop an experimentally feasible protocol to overcome some challenges faced by existing proposals and to probe time-ordered and OTO correlation functions. Our study opens new avenues for both the theoretical and experimental exploration of thermalization and information scrambling dynamics.Here, we study both time-ordered and OTO correlators in a diffusive many-body system by considering the concrete example of the non-integrable, one-dimensional Bose-Hubbard model. Thus far, it is a largely open question, how OTO correlators spread in diffusive systems with a few globally conserved quantities [10,13,15,16]. In our work, we study this question by performing matrix-product operator (MPO) based simulations of the Bose-Hubbard model at high temperatures, at which well defined quasi-particles cease to exist. We demonstrate that in this regime the time-ordered one-particle correlation functions are strongly incoherent and feature rapidly decaying excitations, whereas the OTO correlators indeed describe the ballistic spreading of information across the quantum system (see figure 1). In contrast to the linear light-cone spreading of quantum information, the eventual global thermalization of the closed system takes parametrically longer, due to hydrodynamic power-laws resulting from globally conserved quantities. For example, we show that the local density correlation function decays as Dt 1, describing diffusion in one dimension with the corresponding diffusion constant D. Thus, the time scales associated with the spread of information and with global thermalization are different.Despite their usefulness to characterize interacting many-body systems theoretically, it remains a challenge to experimentally measure such dynamical correlation functions in real space and time [17,18], as required to observe information spreading. Here, we propose generic experimental protocols to characterize both timeordered and OTO correlators via local many-body interferometry. Our proposal to measure OTO correlators is unique because it overcomes some of the challenges that recently proposed pro...

We study the equilibrium time correlations for the conserved fields of classical anharmonic chains and argue that their dynamic correlator can be predicted on the basis of nonlinear fluctuating hydrodynamics. In fact, our scheme is more general and would also cover other one-dimensional Hamiltonian systems, for example, classical and quantum fluids. Fluctuating hydrodynamics is a nonlinear system of conservation laws with noise. For a single mode, it is equivalent to the noisy Burgers equation, for which explicit solutions are available. Our focus is the case of several modes. No exact solution has been found so far, and we rely on a one-loop approximation. The resulting mode-coupling equations have a quadratic memory kernel and describe the time evolving 3×3 correlator matrix of the locally conserved fields. Long time asymptotics is computed analytically, and finite time properties are obtained through a numerical simulation of the mode-coupling equations.

Upon doping, Mott insulators often exhibit symmetry breaking where charge carriers and their spins organize into patterns known as stripes.For high-T c superconducting cuprates, stripes are widely suspected to exist in a fluctuating form. Here, we use numerically exact determinant quantum Monte Carlo calculations to demonstrate dynamical stripe correlations in the three-band Hubbard model, which represents the local electronic structure of the Cu-O plane. Our results, which are robust to varying parameters, cluster size, and boundary condition, strongly support the interpretation of a variety of experimental observations in terms of the physics of fluctuating stripes, including the hourglass magnetic dispersion and the Yamada plot of incommensurability vs. doping. These findings provide a novel perspective on the intertwined orders emerging from the cuprates' normal state. 2Recent experiments have established charge stripes as universal in underdoped cuprate superconductors (1, 2 ). In contrast, no consensus exists regarding the universality of spin stripes, which are present and intimately tied to charge stripes in many doped Mott insulators (1, 3-5 ) but absent, at least in the static long-range form, in the majority of cuprates.Whether spin stripes exist in a more subtle fluctuating form in these cuprates remains an open and controversial question, of importance due to theoretical proposals suggesting a link between fluctuating stripes and the mechanism of high-Tc superconductivity (6-10 ).The evidence for fluctuating spin stripes in the cuprates has revolved around ubiquitous observations of an hourglass-shaped magnetic excitation spectrum (11, 12 ). Its presence both in compounds that exhibit static stripe order (13 ) and in those that do not (14, 15 ) finds a natural explanation in the concept of fluctuating stripes (9, 16 ). However, alternative interpretations based on itinerant electrons exist (17 ) We begin by studying the 16 × 4 rectangular cluster with fully periodic boundary conditions. Figure 1A presents the real space, equal time spin correlation function from our finite temperature DQMC simulations at half-filling. In the undoped state, as in prior studies, copper spin correlations are dominated by commensurate antiferromagnetism, evident through the checkerboard pattern in the spin correlation function or equivalently the uniform phase of the staggered spin correlations. At p = 0.042 hole doping (Fig. 1B) The elevated temperatures where stripe correlations are seen imply surprisingly strong stripe correlations over a substantial doping range. As shown in the supplement, stripe order is robust to different choices of Hubbard model parameters (Fig. S1). Moreover, stripe order persists for larger rectangular clusters (16 × 6, Fig. S2), and additional stripes begin 5 to develop as the transverse dimension increases (8 × 8 and 10 × 10, Fig. S3). This is consistent with DMRG results showing strong stripe tendencies for larger cluster sizes (20, 21 ), indicating that our observations are not artifacts...