Entanglement and matrix elements of observables in interacting integrable systems

TL;DR

This paper investigates entanglement entropy and local operator matrix elements in the eigenstates of an interacting integrable system, revealing distinct features from quantum chaotic systems and similarities with quadratic fermionic models.

Contribution

It provides a detailed comparison of entanglement and matrix elements in integrable versus chaotic systems, highlighting unique entanglement scaling and statistical properties.

Findings

Entanglement entropy has a smaller volume-law coefficient than in chaotic systems.

Eigenstate entanglement entropy closely matches that of quadratic fermionic models.

Diagonal matrix elements support does not vanish with system size, fluctuations decay as a power law.

Abstract

We study the bipartite von Neumann entanglement entropy and matrix elements of local operators in the eigenstates of an interacting integrable Hamiltonian (the paradigmatic spin-1/2 XXZ chain), and we contrast their behavior with that of quantum chaotic systems. We find that the leading term of the average (over all eigenstates in the zero magnetization sector) eigenstate entanglement entropy has a volume-law coefficient that is smaller than the universal (maximal entanglement) one in quantum chaotic systems. This establishes the entanglement entropy as a powerful measure to distinguish integrable models from generic ones. Remarkably, our numerical results suggest that the volume-law coefficient of the average entanglement entropy of eigenstates of the spin-1/2 XXZ Hamiltonian is very close to, or the same as, the one for translationally invariant quadratic fermionic models. We also study matrix elements of local operators in the eigenstates of the spin-1/2 XXZ Hamiltonian at the center of the spectrum. For the diagonal matrix elements, we show evidence that the support does not vanish with increasing system size, while the average eigenstate-to-eigenstate fluctuations vanish in a power-law fashion. For the off-diagonal matrix elements, we show that they follow a distribution that is close to (but not quite) log-normal, and that their variance is a well-defined function of $\omega=E_{\alpha}-E_{\beta}$ ($\{E_{\alpha}\}$ are the eigenenergies) proportional to $1/D$, where $D$ is the Hilbert space dimension.