Reflected entropy for free scalars

Abstract

We continue our study of reflected entropy, $R(A,B)$, for Gaussian systems. In this paper we provide general formulas valid for free scalar fields in arbitrary dimensions. Similarly to the fermionic case, the resulting expressions are fully determined in terms of correlators of the fields, making them amenable to lattice calculations. We apply this to the case of a $(1+1)$-dimensional chiral scalar, whose reflected entropy we compute for two intervals as a function of the cross-ratio, comparing it with previous holographic and free-fermion results. For both types of free theories we find that reflected entropy satisfies the conjectural monotonicity property $R(A,BC) \geq R(A,B)$. Then, we move to $(2+1)$ dimensions and evaluate it for square regions for free scalars, fermions and holography, determining the very-far and very-close regimes and comparing them with their mutual information counterparts. In all cases considered, both for $(1+1)$- and $(2+1)$-dimensional theories, we verify that the general inequality relating both quantities, $R(A,B)\geq I(A,B)$, is satisfied. Our results suggest that for general regions characterized by length-scales $L_A\sim L_B\sim L$ and separated a distance $\ell$, the reflected entropy in the large-separation regime ($x\equiv L/\ell \ll 1$) behaves as $R(x) \sim - I(x) \log x$ for general CFTs in arbitrary dimensions.