Interval, Sphere, and Cylinder Entanglement in CFT

Intervals, spheres, and cylinders are calculable in CFT because conformal maps relate their reduced-state path integrals or modular flow to simpler geometries. The formulas are exact only for the mapped state and global spacetime identification; adding a mass, changing the state, or ignoring the cylinder circumference leaves their domain.

Required background. Use Universal Terms and Entangling-Surface Geometry. Helpful background. Replica and branched geometries supplies the path-integral construction.

One interval on line, circle, and thermal cylinder

For a two-dimensional CFT vacuum on the infinite line,

$$S_A=\frac{c}{3}\log\frac{\ell}{\epsilon}+c_1.$$

On a circle of circumference $L$ in its vacuum,

$$S_A=\frac{c}{3} \log\!\left[ \frac{L}{\pi\epsilon} \sin\!\left(\frac{\pi\ell}{L}\right) \right]+c_1,$$

whereas at inverse temperature $\beta$ on the infinite line,

$$S_A=\frac{c}{3} \log\!\left[ \frac{\beta}{\pi\epsilon} \sinh\!\left(\frac{\pi\ell}{\beta}\right) \right]+c_1.$$

The sine and hyperbolic sine encode different global identifications and states. These results follow from twist-operator transformations, as derived by Calabrese and Cardy 2004, § 3.

The structure figure places these formulas in the geometric branch; disjoint and mixed-state quantities require more data.

Conformal maps fix universal shape dependence for specified states and global geometries. A thermal cylinder, vacuum circle, and infinite line are not interchangeable even when their local metrics are conformally related. Schematic.

The ball and hyperbolic space

The causal development of a ball in a CFT vacuum is conformally mapped to a hyperbolic cylinder. Vacuum modular flow becomes time translation, and the reduced state is unitarily related to a thermal state at a fixed temperature. This converts sphere entanglement into thermal free-energy data and connects its universal term to anomaly or sphere-partition-function information; see Casini, Huerta, and Myers 2011, §§2–3.

The statement uses conformal invariance and the vacuum. A relevant mass deformation or generic excited state does not map to the same thermal density operator. On compact spaces, zero modes, spin structures, and boundary conditions can also modify constants and spectra.

As checks, the interval formulas recover the line result for $\ell\ll L,\beta$, respect $S(\ell)=S(L-\ell)$ for a pure circle vacuum, and become extensive for $\ell\gg\beta$ in the thermal case.

Map failures

The validity figure makes the hidden global choices explicit.

The universal coefficient survives matched conformal transformations, but additive constants, anomaly terms, spin structures, and global identifications require care. Massive theories and generic excited states lie outside the vacuum CFT map. Schematic.

Always state whether the cylinder direction is spatial or Euclidean time, the state preparation, circumference or temperature, boundary conditions, and cutoff transformation.

References

• Calabrese, Pasquale, and John Cardy. “Entanglement Entropy and Quantum Field Theory.” Journal of Statistical Mechanics (2004): P06002. DOI. Open preprint.
• Casini, Horacio, Marina Huerta, and Robert C. Myers. “Towards a Derivation of Holographic Entanglement Entropy.” Journal of High Energy Physics 05 (2011): 036. DOI. Open preprint.