A derivation of the late-time volume law for local operator entanglement

Abstract

Local Operator Entanglement (LOE) has emerged an indicator of quantum chaos in many-body systems. Numerical studies have shown that, in chaotic systems, LOE grows linearly in time and displays a volume-law behavior at late times, scaling proportionally with the number of local degrees of freedom. Despite extensive numerical evidence, complemented by analytical studies in integrable systems, a fully analytical understanding of the emergence of the volume law remains incomplete. In this paper, we contribute toward this goal by deriving a late-time expression for LOE in chaotic systems that exhibits volume-law scaling. Our derivation proceeds by expressing the late-time LOE in the Liouville eigenstate basis and relies on three main assumptions: a higher-order non-resonance condition for the Hamiltonian eigenenergies, the Eigenstate Thermalization Hypothesis (ETH) ansatz for the matrix elements of the initial local operator, and the replacement of Hamiltonian eigenstates with random states in the final expression for LOE. Under these assumptions, we obtain an explicit formula displaying volume-law scaling. Finally, we complement our analytical derivation with numerical simulations of the 1D mixed-field Ising model, testing the resulting formula and exploring the regime of validity of our assumptions.

Figures (29)

• Figure 1: A pictorial representation of the bipartition considered throughout this work for the computation of entanglement in the extended space, $\mathcal{H}\otimes\mathcal{H}'$. We consider the case for which the bipartition is induced from a bipartition within the physical Hilbert space, $\mathcal{H}=\mathcal{H}_A\otimes\mathcal{H}_B$, with dimensions $\mathrm{d}_A$ and $\mathrm{d}_B$, respectively, such that $\mathrm{d}_A\mathrm{d}_B=\mathrm{d}$. This way, each part of the doubled space is divided between $\mathcal{H}\otimes\mathcal{H}'\equiv(\mathcal{H}_{A}\otimes\mathcal{H}'_{A})\otimes(\mathcal{H}_{B}\otimes\mathcal{H}'_{B})$.
• Figure 2: Late-time LOE of the full Hilbert space as a function of the size of subsystem A for $L = 6$ (left) and $L=7$ (right), computed through both numerical exact diagonalization (orange full circles) and analytical Haar average (blue empty squares).
• Figure 3: Error function comparing the simulations based on exact diagonalization with Haar random averaged values of late-time LOE calculation after time average for the complete spectrum of the cases $L=6$ (green circles) and $L=7$ (purple squares). Each panel shows the values at different bipartition sizes, including system A with $1$, $L/4$ and $L/2$ spins.
• Figure 4: $F(\mathcal{O})$ (Equation \ref{['eq:F_explicit_form']}) as a function of the size of subsystem A, computed through both numerical exact diagonalization (orange full circles) and analytical Haar average (blue empty squares). We have set $L=10$, and each panel corresponds to a different size of the energy shell.
• Figure 5: Relative error between the simulations based on ED and Haar random analytics for the term $F(\mathcal{O})$ (Equation \ref{['eq:F_explicit_form']}) as a function of the rescaled dimension of the energy shell $\mathrm{d}_w/\mathrm{d}$, with $L=10$. Each panel corresponds to a different bipartition size. From left to right: $n_A=1$, $L/4$ and $L/2$.