Coarse-grained dynamics of operator and state entanglement

TL;DR

The paper provides a coarse-grained, unified framework for entanglement dynamics in chaotic quantum many-body systems, linking state and operator entanglement through a local production rate $\Gamma(\partial S/\partial x)$ and a dual membrane description with line tension $\mathcal{E}(v)$ tied by a Legendre transform. It identifies key speeds—entanglement speed $v_E$ and butterfly speed $v_B$—and gives constraints such as $\mathcal{E}(v_B)=v_B$ and $\mathcal{E}''(v)\ge 0$, with random circuits and a 1D nonintegrable Ising model used to test and illustrate the theory. The work shows spreading operators are not maximally entangled within their footprint, yielding an expanding-pyramid entanglement profile and a measurable $s_{spread}$ related to $\Gamma$, $v_B$, and $v_E$, supported by extensive numerics. It further extends the picture to higher dimensions via membrane minimization, discusses higher-gradient corrections and the role of trace, and outlines potential low-temperature universality in $\mathcal{E}(v)$ and future directions for both theory and numerics.

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 $Γ(\partial S/\partial x)$ 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.

Figures (22)

• Figure 1: Minimal curve picture in 1+1D. At each point in time the directed curve can be assigned a velocity $v(t)$. Its entanglement "energy" is the integral of a velocity-dependent "line tension", plus a possible contribution from the initial state; see Eq. \ref{['spacetimeformula']}.
• Figure 2: Schematic: entanglement growth functions $\Gamma$ and $\mathcal{E}$ (from the finite $q$ random circuit example in Eqs. \ref{['regularexample']}, \ref{['regularexamplegamma']}).
• Figure 3: Entanglement of the time evolution operator. The entanglement of an operator acting on $L$ spins, for example the time evolution operator (left), is calculated by treating it as a state on $2L$ spins (right). $S_U(x,y,t)$ denotes the entanglement of the subsystem consisting of the shaded spins in the upper right figure.
• Figure 4: The unitary entanglement $S_U(x,y,t)$ is proportional to the line tension $\mathcal{E}(v)$ for a cut with $v=(x-y)/t$ (when $|x-y|\leq v_B t$, and neglecting boundary effects).
• Figure 5: Numerical determination of the entanglement line tension of the nonintegrable Ising model. $\mathcal{E}_\text{eff}$ is defined in Eq. \ref{['eeffdefn']} and is expected to converge as in Eq. \ref{['eq:eveff']} at late times. This data is for a system of size $L=12$. Here ${v\equiv |x-y|/t}$, and $(x+y)/2$ is at the centre of the chain to minimize boundary effects (hence $|x-y|$ is even). See also Fig. \ref{['fig:isingev2']}.