Chaos-protected locality

Published for SISSA by Springer Received: September 22, 2021 Revised: December 1, 2021 Accepted: December 30, 2021 Published: January 17, 2022 Shao-Kai Jian and Brian Swingle Department of Physics, Brandeis University, Waltham, Massachusetts 02453, U.S.A. Condensed Matter Theory Center and Joint Quantum Institute, Department of Physics, University of Maryland, College Park, Maryland 20742, U.S.A. E-mail: , Abstract: Microscopic speed limits that constrain the motion of matter, energy, and information abound in physics, from the “ultimate” speed limit set by light to Lieb-Robinson speed limits in quantum spin systems. In addition to these state-independent speed limits, systems can also be governed by emergent state-dependent speed limits indicating slow dynamics arising, for example, from slow low-energy quasiparticles. Here we describe a different kind of speed limit: a situation where complex information/entanglement spreads rapidly, in a fashion inconsistent with any speed limit, but where simple signals continue to obey an approximate speed limit. If we take the point of view that the motion of simple signals defines the local spacetime geometry of the universe, then the effects we describe show that spacetime locality can be compatible with a high degree of non-local interactions provided these are sufficiently chaotic. With this perspective, we sharpen a puzzle about black holes recently raised by Shor and propose a schematic resolution. Keywords: Black Holes, Random Systems, AdS-CFT Correspondence ArXiv ePrint: 2109.03825 Open Access, c The Authors. Article funded by SCOAP3 . https://doi.org/10.1007/JHEP01(2022)083 JHEP01(2022)083 Chaos-protected locality Contents 1 3 2 The LC model 2.1 Simple signal: two-point correlation function 2.2 Non-local information: out-of-time order correlation function 2.3 Backreaction: a large q study 8 9 11 13 3 Entanglement dynamics after a global quench 3.1 Quench protocol and setup 3.2 Time evolution of Rényi entropy 16 16 20 4 A black hole puzzle 4.1 Shor’s cell model 4.2 Notions of scrambling and a potential puzzle 4.3 Sharp puzzle for AdS black holes 22 22 24 25 5 Discussion 29 A Free Majorana model 29 B Large q effective action of the SYK model 30 C Low energy analysis: an irrelevant perturbation at low energies 32 1 Introduction A basic property of the spacetime geometry of the universe is locality: matter, energy, and information cannot be conveyed from one local point to another distant point instantaneously. Motion from an emitter to a receiver requires a non-zero propagation time equal to the distance between the two divided by the speed light. Suppose, however, that there were interactions in nature that violated this locality rule by directly coupling distant points. Would such interactions necessarily spoil the physics of spacetime locality? In this work, we show that the answer to this question is no. This is achieved by exhibiting a model in which there are totally non-local interactions, yet simple local signals approximately obey a speed-of-light-like speed limit. The standard intuition is that such non-local interactions would be easily detectable since they would permit quantum information to be moved essentially instantaneously. This intuition amounts to an implicit model of the non-local interactions as simple short –1– JHEP01(2022)083 1 Introduction 1.1 Detailed overview 1 For example, we may define simple signals as those that can be created and detected using only a few degrees of freedom at a time (or at least a number not growing with system size). –2– JHEP01(2022)083 cuts, wormholes of a sort, through which, say, photons can easily propagate. Upon further thought, however, the situation is not so clear: would such non-local connections be generically detectable by local observers? Intuitively, if simple signals were scrambled upon passing through a would-be shortcut, then it might be very hard to detect that information was being spread non-locally. In other words, if the local structure of spacetime were defined in terms of the propagation of simple signals (created, manipulated, and detected using local equipment built from relatively small sets of degrees of freedom), then it might be the case that non-local connections would be ineffective at propagating such signals in a detectable form. Hence, simple signals could approximately obey the causal structure of a local spacetime, while more complex (and harder to detect) forms of information and entanglement could spread rapidly in a fashion inconsistent with local causality. We show that it is indeed possible for non-local couplings to respect the locality structure of simple signals by exhibiting a model with the desired physics. More precisely, if the local structure of spacetime is defined using the propagation of simple signals, 1 then this local structure is not strongly modified by the non-local couplings. We further argue that the physics exhibited by this simple model should be generic across a broader class of models. The key physical ingredient in the construction is quantum chaos, hence we term it chaos-protected locality. Our considerations in this work were motivated by certain puzzles in the quantum physics of black holes, but the physics we describe is not restricted to that setting. In the black hole context, we will discuss a seeming conflict, recently highlighted by Shor [1], between the local structure of a black hole spacetime and the rate of entanglement generation by the black hole. We argue that the physical effects described here provide a skeleton for a resolution of that puzzle by showing that rapid non-local entanglement growth can coexist with speed limits for simple signals. The rest of the paper is organized as follows. We first give an overview of chaosprotected locality and some models that realize it in the next subsection. Then in section 2 we fully define a precise model, which is a variant of the Sachdev-Ye-Kitaev (SYK) model [2–4]. We show that simple signals are carried by quasiparticles which travel at a speed limited by microscopic parameters, so that the time for a simple signal to travel through two locations is proportional to the distance. Conversely, non-local signals that cannot be detected by simple equipment, for instance those that are characterized by outof-time order correlations (OTOCs) [2, 5, 6], can spread in a much faster manner. In particular, the time for OTOCs to grow significantly between any two locations are given not by the distance but by the logarithm of the system size. It was recently shown that the OTOC and the entanglement between parts of the system are in general characterized by different time scales [1, 7]. Nevertheless, we show in section 3 that our model is a fast scrambler [8] in the sense that it takes time that scales logarithmically in the system size to establish large entanglement between two unentangled halves. This time scale is the (...truncated)