A global quench is an interesting setting where we can study thermalization of subsystems in a pure state. We investigate entanglement entropy (EE) growth in global quenches in holographic field theories and relate some of its aspects to quantities characterizing chaos. More specifically we obtain four key results: 1. We prove holographic bounds on the entanglement velocity v E and the butterfly effect speed v B that arises in the study of chaos. 2. We obtain the EE as a function of time for large spherical entangling surfaces analytically. We show that the EE is insensitive to the details of the initial state or quench protocol. 3. In a thermofield double state we determine analytically the two-sided mutual information between two large concentric spheres separated in time. 4. We derive a bound on the rate of growth of EE for arbitrary shapes, and develop an expansion for EE at early times. In a companion paper [1], these results are put in the broader context of EE growth in chaotic systems: we relate EE growth to the chaotic spreading of operators, derive bounds on EE at a given time, and compare the holographic results to spin chain numerics and toy models. In this paper, we perform holographic calculations that provide the basis of arguments presented in that paper.
A global quench is an interesting setting where we can study thermalization of subsystems in a pure state. We investigate entanglement entropy (EE) growth in global quenches in holographic field theories and relate some of its aspects to quantities characterizing chaos. More specifically we obtain four key results: 1. We prove holographic bounds on the entanglement velocity vE and the butterfly effect speed vB that arises in the study of chaos. 2. We obtain the EE as a function of time for large spherical entangling surfaces analytically. We show that the EE is insensitive to the details of the initial state or quench protocol. 3. In a thermofield double state we determine analytically the two-sided mutual information between two large concentric spheres separated in time. 4. We derive a bound on the rate of growth of EE for arbitrary shapes, and develop an expansion for EE at early times. In a companion paper, these results are put in the broader context of EE growth in chaotic systems: we relate EE growth to the chaotic spreading of operators, derive bounds on EE at a given time, and compare the holographic results to spin chain numerics and toy models. In this paper, we perform holographic calculations that provide the basis of arguments presented in that paper.
Abstract: We discuss the behaviour of holographic entanglement entropy following a local quench in 2+1 dimensional strongly coupled CFTs. The entanglement generated by the quench propagates along an emergent light-cone, reminiscent of the Lieb-Robinson light-cone propagation of correlations in non-relativistic systems. We find the speed of propagation is bounded from below by the entanglement tsunami velocity obtained earlier for global quenches in holographic systems, and from above by the speed of light. The former is realized for sufficiently broad quenches, while the latter pertains for well localized quenches. The non-universal behavior in the intermediate regime appears to stem from finite-size effects. We also note that the entanglement entropy of subsystems reverts to the equilibrium value exponentially fast, in contrast to a much slower equilibration seen in certain spin models.
We discuss the behaviour of holographic entanglement entropy following a local quench in 2+1 dimensional strongly coupled CFTs. The entanglement generated by the quench propagates along an emergent light-cone, reminiscent of the Lieb-Robinson light-cone propagation of correlations in non-relativistic systems. We find the speed of propagation is bounded from below by the entanglement tsunami velocity obtained earlier for global quenches in holographic systems, and from above by the speed of light. The former is realized for sufficiently broad quenches, while the latter pertains for well localized quenches. The non-universal behavior in the intermediate regime appears to stem from finite-size effects. We also note that the entanglement entropy of subsystems reverts to the equilibrium value exponentially fast, in contrast to a much slower equilibration seen in certain spin models.
A bstract The Ryu-Takayanagi (RT) formula and its interpretation in terms of quantum error correction (QEC) implies an emergent locality for the spread of quantum information in holographic CFTs, where information injected at a point in the boundary theory spreads within a sharp light-cone corresponding to the butterfly velocity. This emergent locality is a necessary condition for the existence of a geometric bulk dual with an RT-like formula for entanglement entropy. In this paper, we use tools from QEC to study the spread of quantum information and the emergence of a sharp light-cone in an analytically tractable model of chaotic dynamics, namely a one-dimensional Brownian SYK chain. We start with an infinite temperature state in this model and inject a qudit at time t = 0 at some point p on the chain. We then explicitly calculate the amount of information of the qudit contained in an interval of length 2 ℓ (centered around p ) at some later time t = T . We find that at strong coupling, this quantity shows a sharp transition as a function of ℓ from near zero to near maximal correlation. The transition occurs at ℓ ~ v B T , with v B being the butterfly velocity. Underlying the emergence of this sharp light-cone is a non-linear generalization of the diffusion equation called the FKPP equation, which admits sharp domain wall solutions at late times and strong coupling. These domain wall solutions can be understood on physical grounds from properties of operator growth in chaotic systems.
We give a detailed theory for the leading coarse-grained dynamics of entanglement entropy of states and of operators in generic short-range interacting quantum many-body systems. This includes operators spreading under Heisenberg time evolution, which we find are much less entangled than "typical" operators of the same spatial support. Extending previous conjectures based on random circuit dynamics, we provide evidence that the leading-order entanglement dynamics of a given chaotic system are determined by a function
E(v), which is model-dependent, but which we argue satisfies certain general constraints. In a minimal membrane picture,
E(v) is the "surface tension" of the membrane and is a function of the membrane's orientation
v in spacetime. For one-dimensional (1D) systems this surface tension is related by a Legendre transformation to an entanglement entropy growth rate
Γ(∂S/∂x) which depends on the spatial "gradient" of the entanglement entropy
S(x,t) across the cut at position
x. We show how to extract the entanglement growth functions numerically in 1D at infinite temperature using the concept of the operator entanglement of the time evolution operator, and we discuss possible universality of
E at low temperatures. Our theoretical ideas are tested against and informed by numerical results for a quantum-chaotic 1D spin Hamiltonian. These results are relevant to the broad class of chaotic many-particle systems or field theories with spatially local interactions, both in 1D and above.