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arXiv:2309.03491 · Journal of High Energy Physics

Holographic renormalized entanglement and entropic c-function

Mitsutoshi Fujita A, Song He, Yuan Sun, Jun Zhang

Published Jan 16, 2024• 4 citations
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Abstract

A bstract We compute holographic entanglement entropy (EE) and the renormalized EE in AdS solitons with gauge potential for various dimensions. The renormalized EE is a cutoff-independent universal component of EE. Via Kaluza-Klein compactification of S 1 and considering the low-energy regime, we deduce the ( d − 1)-dimensional renormalized EE from the odd-dimensional counterpart. This corresponds to the shrinking circle of AdS solitons, probed at large l . The minimal surface transitions from disk to cylinder dominance as l increases. The quantum phase transition occurs at a critical subregion size, with renormalized EE showing non-monotonic behavior around this size. Across dimensions, massive modes decouple at lower energy, while degrees of freedom with Wilson lines contribute at smaller energy scales.

DOI
10.1007/jhep01(2024)079

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A bstract We compute holographic entanglement entropy (EE) and the renormalized EE in AdS solitons with gauge potential for various dimensions. The renormalized EE is a cutoff-independent universal component of EE. Via Kaluza-Klein compactification of S 1 and considering the low-energy regime, we deduce the ( d − 1)-dimensional renormalized EE from the odd-dimensional counterpart. This corresponds to the shrinking circle of AdS solitons, probed at large l . The minimal surface transitions from disk to cylinder dominance as l increases. The quantum phase transition occurs at a critical subregion size, with renormalized EE showing non-monotonic behavior around this size. Across dimensions, massive modes decouple at lower energy, while degrees of freedom with Wilson lines contribute at smaller energy scales.
Holographic entanglement entropy (EE) and its renormalized form in AdS solitons with gauge potential are computed across a range of dimensions. The renormalized EE is identified as a cutoff-independent universal component of EE. By taking into account Kaluza-Klein reduction and the constraints of the low-energy limit, the (d − 1)-dimensional renormalized EE is deduced from the odd-dimensional counterpart. The region associated with the shrinking circle, which is examined at large values of l is indicative of this relationship. Transitions of the minimal surface are from a disk-shaped to a cylindrical configuration as l increases. A quantum phase transition occurs at a critical subregion size, with the renormalized EE exhibiting non-monotonic behavior in the vicinity of this size. In all dimensions, massive modes become decoupled at the low energy limit, whereas degrees of freedom that include Wilson lines influence the behavior at smaller energy scales.
We study two-interval holographic entanglement entropy and entanglement wedge cross section in cutoff AdS. In particular, we investigate phase transitions of them. For two-interval entanglement entropy, the transition point monotonically decreases with a deformation parameter, which means that by the TT deformation the degrees of freedom in subsystems are decreasing. This implies that the effect of the TT deformation can be regarded as the rescaling of the energy scale. We also study entanglement wedge cross section in cutoff AdS, and our result implies that for the entanglement of purification in the TT deformed CFTs phase transition could occur even for fixed subsystems.
A bstract Entanglement entropy has proven to be a powerful tool for probing renormalization group (RG) flows in quantum field theories, with c-functions derived from it serving as candidate measures of the effective number of degrees of freedom. While the monotonicity of such c-functions is well established in many settings, notable exceptions occur in theories with a mass scale. In this work, we investigate entanglement c-functions in the context of holographic RG flows, with a particular focus on flows across dimensions induced by circle compactifications. We argue that in spacetime dimensions d ≥ 4, standard constructions of c-functions, which rely on higher derivatives of the entanglement entropy of either a ball or a cylinder, generically lead to non-monotonic behavior. Working with known dual geometries, we argue that the non-monotonicity stems not from any pathology or curvature singularity, but from a transition in the holographic Ryu-Takayanagi surface. In compactifications from four to three dimensions, we propose a modified construction that restores monotonicity in the infrared, although a fully monotonic ultraviolet extension remains elusive. Furthermore, motivated by entanglement entropy inequalities, we conjecture a bound on the cylinder entanglement c-function, which holds in all our examples.
We compute entanglement entropy for free massive scalar fields in anti-de Sitter (AdS) space. The entangling surface is a minimal surface whose boundary is a sphere at the boundary of AdS. The entropy can be evaluated from the thermal free energy of the fields on a topological black hole by using the replica method. In odd-dimensional AdS, exact expressions of the Rényi entropy S n are obtained for arbitrary n. We also evaluate 1-loop corrections coming from the scalar fields to holographic entanglement entropy. Applying the results, we compute the leading difference of entanglement entropy between two holographic CFTs related by a renormalization group flow triggered by a double trace deformation. The difference is proportional to the shift of a central charge under the flow.
We compute holographic entanglement entropy in two strongly coupled nonlocal field theories: the dipole and the noncommutative deformations of SYM theory. We find that entanglement entropy in the dipole theory follows a volume law for regions smaller than the length scale of nonlocality and has a smooth cross-over to an area law for larger regions. In contrast, in the noncommutative theory the entanglement entropy follows a volume law for up to a critical length scale at which a phase transition to an area law occurs. The critical length scale increases as the UV cutoff is raised, which is indicative of UV/IR mixing and implies that entanglement entropy in the noncommutative theory follows a volume law for arbitrary large regions when the size of the region is fixed as the UV cutoff is removed to infinity. Comparison of behaviour between these two theories allows us to explain the origin of the volume law. Since our holographic duals are not asymptotically AdS, minimal area surfaces used to compute holographic entanglement entropy have novel behaviours near the boundary of the dual spacetime. We discuss implications of our results on the scrambling (thermalization) behaviour of these nonlocal field theories. ar X iv
We study the universal contributions to the entanglement entropy (EE) of 2+12+12+1-dimensional and 3+13+13+1-dimensional holographic conformal field theories (CFTs) on topologically nontrivial manifolds, focusing on tori. The holographic bulk corresponds to anti--de Sitter-soliton geometries. We characterize the properties of these regulator-independent EE terms as a function of both the size of the cylindrical entangling region, and the shape of the torus. In 2+12+12+1 dimensions, in the simple limit where the torus becomes a thin one-dimensional ring, the EE reduces to a shape-independent constant $2\ensuremath{\gamma}$. This is twice the EE obtained by bipartitioning an infinite cylinder into equal halves. We study the renormalization group flow of $\ensuremath{\gamma}$ by defining a renormalized EE that (1) is applicable to general QFTs, (2) resolves the failure of the area law subtraction, and (3) is inspired by the F-theorem. We find that the renormalized $\ensuremath{\gamma}$ decreases monotonically at small coupling when the holographic CFT is deformed by a relevant operator for all allowed scaling dimensions. We also discuss the question of nonuniqueness of such renormalized EEs both in 2+12+12+1 dimensions and 3+13+13+1 dimensions.