Shape Dependence and Entanglement Variations

Shape dependence asks how an entropy changes when the entangling surface is deformed while the state, algebra, and regulator prescription remain fixed. Smooth perturbative variations can expose universal stress-tensor or replica-defect data; they do not determine the entropy of an arbitrary distant or nonsmooth shape.

Required background. Use contact terms in Ward identities, the CFT stress tensor, and Universal Terms and Entangling-Surface Geometry.

Normal deformations and response kernels

Describe a nearby surface by a small normal displacement $\zeta^a(y)$ from a reference surface $\Sigma$. A regulated entropy functional expands as

$$S[\Sigma_\zeta] =S[\Sigma] +\int_\Sigma \zeta^a(y)\,S_a^{(1)}(y) +\frac12\int_\Sigma\!\int_\Sigma \zeta^a(y)\zeta^b(y')S_{ab}^{(2)}(y,y') +O(\zeta^3).$$

The first variation contains stress-tensor or displacement-operator insertions and possible local contact terms. Symmetry makes it vanish for a plane or sphere under many vacuum CFT deformations. The nonlocal part of the second variation is then controlled by stress-tensor two-point data; Faulkner, Leigh, and Parrikar 2016, §§2–3 derive this directly in CFT.

The structure figure locates shape response on the universal-geometry branch.

Shape derivatives probe a neighborhood of one reference surface. Their universal nonlocal kernels may encode $C_T$ or defect data, while local pieces retain regulator and contact-term choices. Schematic.

A deformed sphere

Expand a spherical deformation in harmonics with a small amplitude $\lambda$. Translation and dilation modes are fixed separately; for physical higher harmonics, the leading universal change is quadratic. In CFT examples, the coefficient is proportional to $C_T$ with a positive harmonic weight, consistent with the sphere being a local extremum. Mezei 2015, §§II–IV gives the higher-dimensional deformed-sphere formula in a broad class of CFTs.

This result is perturbative. Extrapolating the quadratic kernel to a cusp, topology change, or order-one displacement is unjustified. A nonsmooth limit can introduce new logarithms governed by corner data rather than the smooth kernel.

The regulator must move covariantly with the chosen geometric prescription. If a coordinate cutoff is held fixed while the physical surface moves, part of the measured variation may be a cutoff displacement. Contact terms are not optional: they enforce Ward identities and can supply local terms required by symmetry.

Controlled use

The lower diagram identifies which choices must be retained in a shape calculation.

The nonlocal second-variation kernel can be universal, but local counterterms and contact terms remain. Moving the cutoff inconsistently, introducing a cusp, or extrapolating beyond small amplitude changes the problem. Schematic.

Report the reference geometry, normal-vector convention, harmonic normalization, deformation amplitude, state, regulator motion, and subtraction. Verify symmetry zero modes before interpreting a quadratic coefficient.

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. Open preprint.
• Mezei, Márk. “Entanglement Entropy across a Deformed Sphere.” Physical Review D 91 (2015): 045038. DOI. Open preprint.