Bounds on corner entanglement in quantum critical states

The entanglement entropy in many gapless quantum systems receives a contribution from corners in the entangling surface in 2+1d. It is characterized by a universal function $a(\theta)$ depending on the opening angle $\theta$, and contains pertinent low energy information. For conformal field theories (CFTs), the leading expansion coefficient in the smooth limit $\theta \to \pi$ yields the stress tensor 2-point function coefficient $C_T$ . Little is known about $a(\theta)$ beyond that limit. Here, we show that the next term in the smooth limit expansion contains information beyond the 2- and 3-point correlators of the stress tensor. We conjecture that it encodes 4-point data, making it much richer. Further, we establish strong constraints on this and higher order smooth-limit coefficients. We also show that $a(\theta)$ is lower-bounded by a non-trivial function multiplied by the central charge $C_T$ , e.g. $a(\pi/2) \geq (\pi^2 \ln 2)C_T /6$. This bound for 90-degree corners is nearly saturated by all known results, including recent numerics for the interacting Wilson-Fisher quantum critical points (QCPs). A bound is also given for the R\'enyi entropies. We illustrate our findings using O(N) QCPs, free boson and Dirac fermion CFTs, strongly coupled holographic ones, and other models. Exact results are also given for Lifshitz quantum critical points, and for conical singularities in 3+1d.Comment: 10 + 8 pages, 6 figures, 1 + 2 tables. v2: refs added, minor change

Holographic torus entanglement and its RG flow

We study the universal contributions to the entanglement entropy (EE) of 2+1d and 3+1d holographic conformal field theories (CFTs) on topologically non-trivial manifolds, focusing on tori. The holographic bulk corresponds to AdS-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+1d, in the simple limit where the torus becomes a thin 1d ring, the EE reduces to a shape-independent constant $2\gamma$. This is twice the EE obtained by bipartitioning an infinite cylinder into equal halves. We study the RG flow of $\gamma$ 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 $\gamma$ decreases monotonically when the holographic CFT is deformed by a relevant operator for all allowed scaling dimensions. We also discuss the question of non-uniqueness of such renormalized EEs both in 2+1d and 3+1d.Comment: 22 pages, 11 figures, v2: minor changes, refs. adde

Quantum critical response: from conformal perturbation theory to holography

We discuss dynamical response functions near quantum critical points, allowing for both a finite temperature and detuning by a relevant operator. When the quantum critical point is described by a conformal field theory (CFT), conformal perturbation theory and the operator product expansion can be used to fix the first few leading terms at high frequencies. Knowledge of the high frequency response allows us then to derive non-perturbative sum rules. We show, via explicit computations, how holography recovers the general results of CFT, and the associated sum rules, for any holographic field theory with a conformal UV completion -- regardless of any possible new ordering and/or scaling physics in the IR. We numerically obtain holographic response functions at all frequencies, allowing us to probe the breakdown of the asymptotic high-frequency regime. Finally, we show that high frequency response functions in holographic Lifshitz theories are quite similar to their conformal counterparts, even though they are not strongly constrained by symmetry.Comment: 45+14 pages, 9 figures. v2: small clarifications, added reference