Reflected entropy for free scalars
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Springer
Received: September 11, 2020 Accepted: October 24, 2020 Published: November 26, 2020
Reflected entropy for free scalars
Instituto Balseiro, Centro Atómico Bariloche, 8400-S.C. de Bariloche, Río Negro, Argentina
E-mail: [email protected], [email protected] 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) ≥ 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) ≥ I(A, B), is satisfied. Our results suggest that for general regions characterized by length-scales LA ∼ LB ∼ L and separated a distance `, the reflected entropy in the large-separation regime (x ≡ L/` 1) behaves as R(x) ∼ −I(x) log x for general CFTs in arbitrary dimensions. Keywords: Conformal Field Theory, AdS-CFT Correspondence, Field Theories in Lower Dimensions ArXiv ePrint: 2008.11373
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP11(2020)148
JHEP11(2020)148
Pablo Bueno and Horacio Casini
Contents 1
2 Reflected entropy for free scalars 2.1 Purification and general formulas: take one 2.2 Purification and general formulas: take two
4 4 7
3 Reflected entropy for a d = 2 chiral scalar 3.1 Reflected entropy for two intervals 3.2 Eigenvalues spectrum 3.3 Monotonicity of reflected entropy
9 9 12 15
4 Reflected entropy in d = 3 4.1 Free scalar correlators 4.2 Free fermion correlators 4.3 Reflected entropy for two parallel squares
16 16 17 18
5 Outlook
23
A Numerical values of Rferm. /c and Rscal. /c
24
1
Introduction
Entanglement entropy (EE) of subregions is an ill-defined quantity in quantum field theory (QFT). This fact can be understood from various perspectives. From a lattice point of view, as we reduce the lattice spacing a growing amount of entanglement across the entangling surface adds up, producing the usual area-law divergence (and others) in the limit. From the continuum theory perspective, the underlying reason has to do with the fact that algebras of operators associated to spatial regions are von Neumann algebras of type-III, for which all traces are either vanishing or infinite — see e.g. [1, 2]. The situation impro
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