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arXiv:1803.08627 · Physical review. D/Physical review. D.

Holographic complexity of the disk subregion in ( 2+1 )-dimensional gapped systems

Lin-Peng Du, Shao-Feng Wu, Hua-Bi Zeng

Published Sep 14, 2018• 9 citations
arXivHTMLPDF

Abstract

Using the volume of the space enclosed by the Ryu-Takayanagi (RT) surface, we study the complexity of the disk-shape subregion (with radius R) in various ($2+1$)-dimensional gapped systems with gravity dual. These systems include a class of toy models with singular IR and the bottom-up models for quantum chromodynamics and fractional quantum Hall effects. Two main results are: (i) in the large-R expansion of the complexity, the R-linear term is always absent, similar to the absence of topological entanglement entropy, and (ii) when the entanglement entropy exhibits the classic ``swallowtail'' phase transition, the complexity is sensitive but reacts differently.

DOI
10.1103/physrevd.98.066005

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Entanglement entropy in holographic conformal field theories with boundaries of generic spacetime signature cannot be described entirely within the standard holographic duality to a real bulk spacetime. We show, both in static and time-dependent setups, that an exact match with the conformal field theory predictions in two boundary dimensions entails an extension of the dual spacetime to complex coordinates. As a consequence, relevant bulk objects such as Ryu-Takayanagi surfaces and branes are in general complex, yet anchored on a real asymptotic boundary preserving the physical meaning of field-theoretical quantities. We also provide a third numerical check of the holographic prediction with a two-dimensional system of free Gaussian fermions exhibiting the same asymptotic regimes across different entanglement phases.
We check formally that the Hubeny-Rangamani-Takayanagi prescription for holographic entanglement entropy---when applied to a static black brane spacetime and to a wide class of subregions that do not lie on a constant time slice---gives rise to volume-law entropy in the limit of large subregion. By volume-law entropy, we mean that the entanglement entropy scales with the volume of the projection of the boundary subregion onto a static time slice with respect to the boundary thermal state (with the same coefficient as in the volume law on the static time slice). Our result applies to subregions that have reflection symmetry as well as strips, and we also present field-theoretic arguments in support of our holographic findings.