Parameter dependence of entanglement spectra in quantum field theories

Abstract

In this paper, we explore the characteristics of reduced density matrix spectra in quantum field theories. Previous studies mainly focus on the function $\mathcal{P}(λ):=\sum_i δ(λ-λ_i)$, where $λ_i$ denote the eigenvalues of the reduced density matirx. We introduce a series of functions designed to capture the parameter dependencies of these spectra. These functions encompass information regarding the derivatives of eigenvalues concerning the parameters, notably including the function $\mathcal{P}_{α_J}(λ):=\sum_i \frac{\partial λ_i }{\partial α_J}δ(λ-λ_i)$, where $α_J$ denotes the specific parameter. Computation of these functions is achievable through the utilization of Rényi entropy. Intriguingly, we uncover compelling relationships among these functions and demonstrate their utility in constructing the eigenvalues of reduced density matrices for select cases. We perform computations of these functions across several illustrative examples. Specially, we conducted a detailed study of the variations of $\mathcal{P}(λ)$ and $\mathcal{P}_{α_J}(λ)$ under general perturbation, elucidating their physical implications. In the context of holographic theory, we ascertain that the zero point of the function $\mathcal{P}_{α_J}(λ)$ possesses universality, determined as $λ_0=e^{-S}$, where $S$ denotes the entanglement entropy of the reduced density matrix. Furthermore, we exhibit potential applications of these functions in analyzing the properties of entanglement entropy.