Reflected entropy, symmetries and free fermions

TL;DR

The paper develops a general framework for reflected entropy in quantum field theories with symmetries by introducing a type-I entropy for orbifold (neutral) subalgebras and relating it to the full reflected entropy via a twist-entropy term, $R_{ m F}-S^{\rm I}_{\mathcal O}=\tfrac{1}{2}S_{\tau}$. It then specializes to Gaussian fermions, showing that reflected entropy can be computed from two-point correlators, and performs detailed lattice and continuum analyses for two intervals in a 2D massless Dirac field, including a comparison with holographic results. The authors construct explicit twist operators for ${\mathbb Z}_2$ and ${\rm U}(1)$ symmetries, deriving their correlator-based expectation values and using them to obtain the type-I entropy for the bosonic subalgebra, $S^{\rm I}_{\rm bos.}$. Overall, the work provides practical methods for finite, regulator-independent entropies in QFT with symmetries and opens paths to higher-dimensional and holographic connections, along with insights into the spatial distribution of the relevant type-I algebras.

Abstract

Exploiting the split property of quantum field theories (QFTs), a notion of von Neumann entropy associated to pairs of spatial subregions has been recently proposed both in the holographic context -- where it has been argued to be related to the entanglement wedge cross section -- and for general QFTs. We argue that the definition of this "reflected entropy" can be canonically generalized in a way which is particularly suitable for orbifold theories -- those obtained by restricting the full algebra of operators to those which are neutral under a global symmetry group. This turns out to be given by the full-theory reflected entropy minus an entropy associated to the expectation value of the "twist" operator implementing the symmetry operation. Then we show that the reflected entropy for Gaussian fermion systems can be simply written in terms of correlation functions and we evaluate it numerically for two intervals in the case of a two-dimensional Dirac field as a function of the conformal cross-ratio. Finally, we explain how the aforementioned twist operators can be constructed and we compute the corresponding expectation value and reflected entropy numerically in the case of the $\mathbb{Z}_2$ bosonic subalgebra of the Dirac field.

Figures (6)

• Figure 1: Reflected entropy normalized by the central charge, $R/c$, as a function of the conformal cross-ratio $\eta$ for holographic Einstein gravity (black) and a free fermion (red line and dots). The gray dashed line corresponds to the general-theory behavior for $\eta \rightarrow 1$. For $\eta=1/2$ the holographic result undergoes a phase-transition and the leading $N$ term drops to zero for smaller values of the cross ratio.
• Figure 2: We plot the "leading" eigenvalues of $D|_A$ and $C_{AA^*}$ (as defined in the main text) corresponding, respectively to: the correlators matrix required for the evaluation of the usual type-III entanglement entropy for a single interval and the reflected entropy $R(A,B)$ for a fixed value of the cross-ratio $\eta=25/36$, for different numbers of lattice points. The plot is logarithmic to make the behavior of the different eigenvalues more visible.
• Figure 3: Spatial density $d(x)$ for the type-I factor ${\cal N}_{AB}$ (black curve and red lines). For $x \in A$, $d(x)=1$, whereas for $x\in B$, $d(x)=0$. Between the two intervals, the density interpolates continuously. As $|x|\rightarrow \infty$, $d(x)\rightarrow 1/2$. The mirrored gray curve (plus the pale blue lines) corresponds to the density of ${\cal N}'_{AB}$ --- see eq. (\ref{['tomo']}) for definitions.
• Figure 4: (Left) Expectation value of the standard twist operator $\tau_{\theta}$ for $\theta=\pi/4, \pi/2,\pi$ as a function of the cross-ratio $\eta$ for a free chiral fermion. (Right) Twist entropy $S_{\tau}$ associated to the $\mathbb{Z}_2$ symmetry, as defined in eq. (\ref{['z22']}), for a free chiral fermion. The curve continuously grows from $S_{\tau}=0$ at $\eta=0$ to $S_{\tau}=\log 2$, its maximum value, at $\eta=1$.
• Figure 5: (Left) We plot the reflected entropy for the full fermion algebra, $R_{\rm ferm.}$, the type-I entropy for the bosonic subalgebra, $S^{\rm I}_{\rm bos.}$, and the mutual information for the fermion $I_{\rm ferm.}$ as a function of the cross-ratio $\eta$. (Right) We plot the free-fermion and free-scalar mutual informations, $I_{\rm ferm.}$ and $I_{\rm scal.}$ as well as $I_{\rm ferm.}-S_{\tau}$.