Abstract
We consider the holographic candidate for the entanglement of purification EP, given by the minimal cross sectional area of an entanglement wedge EW. The EP is generally very complicated quantity to obtain in field theories, thus to establish the conjectured relationship one needs to test if EW and EP share common features. In this paper the entangling regions we consider are slabs, concentric spheres, and creases in field theories in Minkowski space. The latter two can be mapped to regions in field theories defined on spheres, thus corresponding to entangled caps and orange slices, respectively. We work in general dimensions and for slabs we also consider field theories at finite temperature and confining theories. We find that EW is neither a monotonic nor continuous function of a scale. We also study a full ten-dimensional string theory geometry dual to a non-trivial RG flow of a three-dimensional Chern-Simons matter theory coupled to fundamentals. We show that also in this case EW behaves non-trivially, which if connected to EP, lends further support that the system can undergo purification simply by expansion or reduction in scale.
Highlights
We study a full ten-dimensional string theory geometry dual to a non-trivial RG flow of a three-dimensional Chern-Simons matter theory coupled to fundamentals
We find that the entanglement wedge cross section is not monotonic under RG flow, or equivalently, at different length scales in the following sense
Symmetry of the strip configuration and bulk metric still impose that, assuming that the entanglement wedge is connected, the entanglement wedge cross section is given by the area of a constant-x hypersurface, Γ, located in the middle of the strips
Summary
Consider two subsystems of the boundary, A and B with no non-zero overlap. Let ΓmABin denote the minimal RT-surface associated with the union AB. Symmetry of the strip configuration and bulk metric still impose that, assuming that the entanglement wedge is connected, the entanglement wedge cross section is given by the area of a constant-x hypersurface, Γ, located in the middle of the strips. This time the induced metric on Γ is (3.8). One important difference between this black hole background and the pure AdSd+1 of previous section is that there exists a critical separation sc such that if s > sc, no connected phase exists for any l [26]. We will find that, in general, neither the mutual information I nor the entanglement wedge cross section EW are monotonic, but show structure at the underlying length scales
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