Real-time correlators in chaotic quantum many-body systems

Abstract

We study real-time correlators $\ensuremath{\langle}\mathcal{O}(\mathbf{x},t)\mathcal{O}(0,0)\ensuremath{\rangle}$ of local operators in chaotic quantum many-body systems. These correlators show universal structure at late times, determined by the geometry of the dominant operator-space Feynman trajectories for the evolving operator $\mathcal{O}(\mathbit{x},t)$. These trajectories, which involve the operator contracting to a point at both the initial and final time, are qualitatively different from those that dominate the out-of-time-order correlation function. In the absence of conservation laws, local correlations decay exponentially: $\ensuremath{\langle}\mathcal{O}(\mathbf{x},t)\mathcal{O}(0,0)\ensuremath{\rangle}\ensuremath{\sim}exp(\ensuremath{-}{s}_{\text{eq}}\phantom{\rule{0.16em}{0ex}}r(\mathbit{v})\phantom{\rule{0.16em}{0ex}}t)$, where $\mathbit{v}=\mathbit{x}/t$ defines a ray in spacetime, and $r(\mathbit{v})$ is a rate function associated with this ray. We express $r(\mathbit{v})$ in terms of cost functions for various spacetime structures. In $1+1\mathrm{D}$ the operator histories can exhibit a phase transition at a critical value ${v}_{c}$ of the ray velocity, which leads to a singular behavior in $r(\mathbit{v})$. At low velocities, the dominant Feynman histories are ``fat'': The operator grows to a size of order ${t}^{\ensuremath{\alpha}}$ (with $\ensuremath{\alpha}=1/2$ in the simplest case) before contracting to a point again. At high velocities the trajectories are ``thin'': The operator always remains of order-one size. In a Haar-random quantum circuit, this transition maps to a simple binding transition for a pair of random walks, which represent the left and right spatial boundaries of the operator. In higher-dimensional systems, thin trajectories always dominate. We discuss the circumstances in which the butterfly velocity ${v}_{B}$ can be deduced from a time-ordered two-point function, rather than the out-of-time ordered correlator. In the random circuit, correlators may also be computed in the framework of an effective Ising-like statistical mechanics model: we describe this calculation, as well as a special feature of the weights in the case of a $1+1\mathrm{D}$ Haar-random brickwork circuit. The present paper addresses lattice models, but also suggests the possibility of morphological phase transitions for real-time Feynman diagrams in continuum quantum field theories.