Abstract
We study the entanglement entropy of free fermions in 2d in the presence of a partially transmitting interface that splits Minkowski space into two half-spaces. We focus on the case of a single interval that straddles the defect, and compute its entanglement entropy in three limits: Perturbing away from the fully transmitting and fully reflecting cases, and perturbing in the amount of asymmetry of the interval about the defect. Using these results within the setup of the Poincaré patch of AdS_22 statically coupled to a zero temperature flat space bath, we calculate the effect of a partially transmitting AdS_22 boundary on the location of the entanglement island region. The partially transmitting boundary is a toy model for black hole graybody factors. Our results indicate that the entanglement island region behaves in a monotonic fashion as a function of the transmission/reflection coefficient at the interface.
Highlights
We extend this repertoire of results by computing the entanglement entropy of 2d free fermions in the presence of a semitransparent defect, but allowing for the region under consideration to be asymmetric about the defect
Recent progress in the black hole information paradox has involved the semiclassical computation of the von Neumann entropy of Hawking radiation, reproducing the Page curve [16, 17]
It is hard to include graybody factors in the computations of the von Neumann entropy of bulk matter fields, and in the setups of [17, 20], which couple a flat space bath to AdS space, the interface is taken to be fully transparent. This is rightly so, since graybody factors are not expected to affect the qualitative features of the Page curve
Summary
Entanglement entropy is an important quantity not just for finite or discrete quantum systems [1] and for continuum quantum field theories and holography [2, 3]. In [8], the intervals that were considered were symmetric about a semitransparent defect that separates 2d flat space into two half-spaces We extend this repertoire of results by computing the entanglement entropy of 2d free fermions in the presence of a semitransparent defect, but allowing for the region under consideration to be asymmetric about the defect. In order to obtain these results, we have used the method of [5] and used the results in [9,10,11] for free fermion determinants in the presence of a boundary with non-translation invariant boundary conditions This method can be used to obtain the entropies of any region, and most of our formal setup carries over to the more general case, but we focus on the particular application at hand to obtain explicit results. For example, [15] for a study of the boundary entropy in holography
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