Real-time correlators in chaotic quantum many-body systems

Abstract

We study real-time local correlators O(x, t)O(0, 0) in chaotic quantum many-body systems. These correlators show universal structure at late times, determined by the dominant operatorspace Feynman trajectories for the evolving operator O(x, t). The relevant trajectories involve the operator contracting to a point at both the initial and final time and so are structurally different from those dominating the out-of-time-order correlator. In the absence of conservation laws, correlations decay exponentially: O(x, t)O(0, 0) ∼ exp(−seqr(v)t), where v = x/t defines a spacetime ray, and r(v) is an associated decay rate. We express r(v) in terms of cost functions for various spacetime structures. In 1+1D, operator histories can show a phase transition at a critical ray velocity vc, where r(v) is nonanalytic. At low v, the dominant Feynman histories are "fat": the operator grows to a size of order t α 1 before contracting to a point again. At high v the trajectories are "thin": the operator always remains of order-one size. In a Haar-random unitary circuit, this transition maps to a simple binding transition for a pair of random walks (the two spatial boundaries of the operator). In higher dimensions, thin trajectories always dominate. We discuss ways to extract the butterfly velocity vB from the time-ordered correlator, rather than the OTOC. Correlators in the random circuit may alternatively be computed with an effective Ising-like model: a special feature of the Ising weights for the Haar brickwork circuit gives vc = vB. This work addresses lattice models, but also suggests the possibility of morphological phase transitions for real-time Feynman diagrams in quantum field theories. CONTENTS B. Noisy spin-1/2 chain C. Brownian circuit VIII. Non-random Floquet and Hamiltonian systems A. Bound phase B. Conserved densities C. Unbound phase IX. Outlook Acknowledgments A. Review of transition probabilities B. Transfer matrix for walks C. Boundary effects in Haar circuit D. Thin cluster approximation in d = 2 E. Brownian circuit: cluster and Ising pictures F. Data for edge-edge correlations G. Boundary conditions in Ising mapping H. Sign phase transition for G(x, t) in d > 1

Random Quantum Circuits

“…4a. By contrast, timeordered correlators such as G(r, t) ≡ Tr[O(r, t)O(0, 0)] are determined by atypical operator trajectories, and look quite different (185): instead of growing ballistically, the string is small at the final time Fig. 4b and c.…”

“…However, there is nontrivial structure in the decay rate r(v) (185). In the 1+1D Haar circuit, the dominant contribution to the correlation function changes from trajectories of the endpoints which are (i) bound together when v > vB to (ii) unbound when v < vB.…”

Random Quantum Circuits

“…In (1+1)-dimensions, typical operator trajectories look like Figure 4a. By contrast, timeordered correlators such as G(r, t ) ≡ Tr[O(r, t )O(0, 0)] are determined by atypical operator trajectories and look quite different (196). Instead of growing ballistically, the string is small at the final time as in Figure 4b,c.…”

“…However, there is nontrivial structure in the decay rate r(v) (196). In the 1+1D Haar circuit, the dominant contribution to the correlation function changes from trajectories of the endpoints, which are (a) bound together when v > v B to (b) unbound when v < v B .…”

“…However, random dynamics can also be formulated in the continuous time limit. An example is the Brownian circuit (98,99,140,196,248,249): This is a Hamiltonian spin chain in which the couplings fluctuate like white noise. Operator spreading in continuoustime noisy models ( 250) is qualitatively similar to what we have discussed in Section 3.2.…”