Corners, Cusps, and Defect Data

Corners and cusps add universal angle-dependent terms that are absent for a smooth entangling surface. Their coefficients carry continuum data only after local perimeter terms, angle conventions, dimension, and regulator anisotropy are controlled.

Required background. Use Universal Terms and Entangling-Surface Geometry and Shape Dependence and Entanglement Variations. Helpful background. Twist and replica defects explain the defect interpretation.

The corner function

For a spatial region with opening angle $\theta$ in a three-dimensional QFT vacuum,

$$S(A) =\alpha\frac{L}{\epsilon} -a(\theta)\log\frac{L}{\epsilon} +O(1).$$

The area-law coefficient $\alpha$ is regulator dependent, while the corner function $a(\theta)$ is universal for the specified theory and angle convention. Purity gives $a(\theta)=a(2\pi-\theta)$. Smoothness requires $a(\pi)=0$ and

$$a(\theta)=\sigma(\pi-\theta)^2+O((\pi-\theta)^4)$$

near $\theta=\pi$. In the sharp limit, $a(\theta)\sim\kappa/\theta$. These limits test numerical extractions over the full angle range.

The chapter map identifies corner data as a specialized geometric branch.

An angle-dependent logarithm isolates data not present for a smooth surface. Its normalization, opening-angle convention, spacetime dimension, and replica-defect realization must be fixed before comparison. Schematic.

Smooth and sharp limits

In a three-dimensional CFT, the smooth-corner coefficient satisfies

$$\frac{\sigma}{C_T}=\frac{\pi^2}{24}$$

with a standard normalization of the stress-tensor two-point function. The relation follows from the universal shape kernel; Faulkner, Leigh, and Parrikar 2016, §§4–5 prove it in field theory. The full function $a(\theta)$ is not determined by $C_T$ alone.

A reliable computation samples several angles, extrapolates at multiple lattice spacings, subtracts the smooth perimeter contribution, and checks both $\theta\to\pi$ and $\theta\to0$. Rotating the same geometric corner relative to an anisotropic lattice is an essential regulator test.

Higher-dimensional conical singularities can generate powers of logarithms and additional local curvature structures. One must not import the three-dimensional $a(\theta)$ formula across dimensions.

Regulator and defect checks

The validity figure shows why a visually identical corner can represent different continuum questions.

Corner logarithms can be universal after perimeter subtraction, but finite-angle discretization, anisotropic cutoffs, interfaces, and defect normalizations remain choices. A coefficient from one dimension or singularity class does not transfer automatically to another. Schematic.

State the interior-angle convention, regulator orientation, fit window, subtraction, $C_T$ normalization, and whether the quantity is a von Neumann or Rényi corner function.

References

• Faulkner, Thomas, Robert G. Leigh, and Onkar Parrikar. “Shape Dependence of Entanglement Entropy in Conformal Field Theories.” Journal of High Energy Physics 04 (2016): 088. DOI.

Further reading

• Bueno, Pablo, Robert C. Myers, and Witold Witczak-Krempa. “Universality of Corner Entanglement in Conformal Field Theories.” Physical Review Letters 115 (2015): 021602. DOI. Open preprint.
• Casini, Horacio, and Marina Huerta. “Universal Terms for the Entanglement Entropy in 2+1 Dimensions.” Nuclear Physics B 764 (2007): 183–201. DOI. Open preprint.