Coarse-grained dynamics of operator and state entanglement

TL;DR

This paper develops a coarse-grained theoretical framework for understanding the entanglement dynamics of states and operators in generic short-range interacting quantum many-body systems, emphasizing a model-dependent entanglement growth function.

Contribution

It introduces a membrane picture with a model-dependent surface tension function $\\mathcal{E}(v)$ that governs entanglement dynamics, extending previous conjectures and providing numerical methods for extraction.

Findings

Entanglement spreading of operators is less than typical operators.

The entanglement growth rate is related to a surface tension function.

Numerical methods for extracting entanglement functions in 1D systems are proposed.

Abstract

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 $\mathcal{E}(v)$, which is model-dependent, but which we argue satisfies certain general constraints. In a minimal membrane picture, $\mathcal{E}(v)$ 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 $\Gamma(\partial S/\partial x)$ 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 $\mathcal{E}$ 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.