Real-time correlators in chaotic quantum many-body systems

Abstract

We study real-time correlators (O(x,t)O(0,0)) 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 O(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: (O(x,t)O(0,0))∼exp(-seqr(v)t), where v=x/t defines a ray in spacetime, and r(v) is a rate function associated with this ray. We express r(v) in terms of cost functions for various spacetime structures. In 1+1D the operator histories can exhibit a phase transition at a critical value vc of the ray velocity, which leads to a singular behavior in r(v). At low velocities, the dominant Feynman histories are "fat": The operator grows to a size of order tα (with α=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 vB 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+1D 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.