Bounds on corner entanglement in quantum critical states

Abstract

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(θ) depending on the opening angle θ, and contains pertinent low energy information. For conformal field theories (CFTs), the leading expansion coefficient in the smooth limit θ → π yields the stress tensor 2-point function coefficient CT . Little is known about a(θ) 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(θ) is lowerbounded by a non-trivial function multiplied by the central charge CT , e.g. a(π/2) ≥ (π 2 log 2)CT /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 Rényi 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.

Universal corner entanglement of Dirac fermions and gapless bosons from the continuum to the lattice

“…It was further found that even at θ = π/2 (away from the smooth limit), a 1 (π/2)/C T takes approximately the same value in a wide range of CFTs. 26,27,38 Specifically, a 1 (π/2)/C T ≈ 1.2, which nearly saturates the lower bound of (π 2 /6) ln 2 ≈ 1.14, Eq. (7).…”

“…As advertised, a (M ) is determined by the first M coefficients of the smooth limit expansion of a(θ). Using recently derived bounds 38 for the coefficients σ (p) , we can explicitly show that a…”

“…α , we need to solve an initial value problem by providing M initial conditions at θ = π, 38 namely ∂…”

“…All theories for which some or all of the σ (p) α are known satisfy these positivity and analyticity properties. 38 …”

Holographic torus entanglement and its renormalization group flow

“…We emphasize the fact that it is the same constant κ that appears in both divergences because the small-angle corner coefficient is also directly related to the EE of a thin strip [39,41]. It would be interesting to see if other connections exist between a(θ) and χ(θ), such as some of the bounds that apply to a(θ) [48]. In d > 3, one can also make connections between χ and higher dimensional analogues of a(θ), such as the coefficient associated with cones in 4d, however we shall not describe them here.…”