Parameter dependence of entanglement spectra in quantum field theories

In this paper, we explore the characteristics of reduced density matrix spectra in quantum field theories. Previous studies mainly focus on the function $\mathcal{P}(\lambda )≔{\sum }_i\delta (\lambda -{\lambda }_i)$, where ${\lambda }_i$ denote the eigenvalues of the reduced density matrix. 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 ${\mathcal{P}}_{{\alpha }_J}(\lambda )≔{\sum }_i\frac{\partial {\lambda }_i}{\partial {\alpha }_J}\delta (\lambda -{\lambda }_i)$, where ${\alpha }_J$ 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. Especially, we conducted a detailed study of the variations of $\mathcal{P}(\lambda )$ and ${\mathcal{P}}_{{\alpha }_J}(\lambda )$ under general perturbation, elucidating their physical implications. In the context of holographic theory, we ascertain that the zero point of the function ${\mathcal{P}}_{{\alpha }_J}(\lambda )$ possesses universality, determined as ${\lambda }_0=e^{-S}$, 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.

Figure 1

The illustration of $\mathcal{P}(\lambda )$ and ${\mathcal{P}}_L(\lambda )$ in the case of the vacuum state of 2D CFT, where the arrows represent the Dirac delta function.

Figure 2

The plots of ${\mathcal{P}}_L(\lambda )$ with various parameters in the scenario of a single interval on a cylinder. We omit the representation of the term $\delta ({\lambda }_m-\lambda )$ in the plot, as it is not important for our current discussion.

Figure 3

Illustration of the function $\mathcal{P}(\lambda )$ in the perturbation state. The black line is the unperturbed function ${\mathcal{P}}_0$. The range of the eigenvalue would change under perturbation. The red line illustrates the perturbation of the function $\mathcal{P}$ due to the alteration in the range of the variable $\lambda$. But it does not satisfy the normalization condition ${\int }_0^{{\lambda }_m}d\lambda \mathcal{P}(\lambda )\lambda =1$. The blue one includes the adjustment to satisfy the normalization.

Figure 4

Illustration of the zero point $\lambda$ varies with dimension $d$.

Figure 5

Illustration of the function ${\mathcal{P}}_L$. The black line and red line are two typical functions for ${\mathcal{P}}_L$. ${\lambda }_0$ is the zero point of ${\mathcal{P}}_L$, and ${\lambda }_m$ is the maximal eigenvalue.

Figure 6

(a) Shows two types of minimal surfaces ${\gamma }_A$ and ${\gamma }_{A^c}\cup {\gamma }_{\mathrm{BH}}$, which are shown in red and purple lines, respectively. We notice that ${\gamma }_A$ is homotopy with ${\gamma }_{A^c}\cup {\gamma }_{\mathrm{BH}}$ instead of ${\gamma }_{A^c}$ in this case. Since the result should take the minimum value in the extreme surfaces, for $L<L_{\mathrm{c}}$ the $S$ is given by ${\gamma }_A$, whereas for $L>L_{\mathrm{c}}$ the $S$ is given by ${\gamma }_{A^c}\cup {\gamma }_{\mathrm{BH}}$, which are shown in red and purple, respectively, in (b).