Entanglement Negativity in Flat Holography

Abstract

We advance holographic constructions for the entanglement negativity of bipartite states in a class of (1+1)-dimensional Galilean conformal field theories dual to asymptotically flat three dimensional bulk geometries described by Einstein Gravity and Topologically Massive Gravity. The construction involves specific algebraic sums of the lengths of bulk extremal curves homologous to certain combinations of the intervals appropriate to such bipartite states. Our analysis exactly reproduces the corresponding replica technique results in the large central charge limit. We substantiate our construction through a semi classical analysis involving the geometric monodromy technique for the case of two disjoint intervals in such holographic Galilean conformal field theories.

Entanglement wedge in flat holography and entanglement negativity

“…In this article, we address this outstanding issue and establish a novel construction for the bulk entanglement wedge and provide a definition for the EWCS in bulk asymptotically flat (2+1)-dimensional Einstein gravity and TMG dual to GC F T 1+1 s. Subsequently we compute the holographic entanglement negativity for various bipartite states in such GC F T 1+1 s through the EWCS generalizing the construction of [35,36] to the framework of flat holography. Remarkably the holographic entanglement negativity computed through our construction matches up to an additive constant with the results described in [64] from the alternative proposal involving the algebraic sum of the lengths of specific combinations of bulk extremal curves as well as with the field theory replica technique results described in [56] in the large central charge limit. This provides an extremely strong substantiation and consistency check for our construction of the bulk EWCS and the consequent computation for the entanglement negativity in the context of flat holography.…”

“…This establishes the construction of the HRT-like extremal curves [61,64] homologous to an interval in GC F T 1+1 s dual to (2+1)-dimensional asymptotically flat bulk theories described by Einstein gravity. We will use some of the results mentioned above in our computation of the EWCS in section 4.…”

“…Finite sized version of this computation, as performed in [64], involves a GC F T 1+1 compactified on a spatial circle of circumference L φ . The bulk dual geometry is the global Minkowski orbifold, which is a quotient of the usual Minkowski spacetime with a compact spatial circle (u, φ) ∼ (u, φ + L φ ), with the metric [60,61]…”

“…The length of the extremal curve homologous to an interval A in the dual GC F T 1+1 may then be obtained following a similar procedure as the previous cases to be [64] L extr total =…”

“…In this context, given the developments described above, a holographic description of mixed state entanglement in such GC F T 1+1 s dual to asymptotically flat bulk geometries was a significant open issue. This interesting issue was investigated recently in [64] where an elegant and clear holographic characterization for the entanglement negativity of bipartite states in a class of GC F T 1+1 s dual to bulk asymptotically flat (2+1)-dimensional Einstein gravity and topologically massive gravity (TMG) was established. Their construction involved specific algebraic sums of the lengths of extremal curves (HRT like surfaces termed swing surfaces [65]) homologous to certain combinations of intervals appropriate to the bipartite state in question.…”