Universal Terms and Entangling-Surface Geometry

Entanglement entropy contains local ultraviolet divergences and, in suitable dimensions and geometries, universal logarithmic or finite terms. A coefficient is universal only after the state, spacetime dimension, entangling geometry, algebra, and allowed counterterms are fixed; not every cutoff-independent-looking constant survives a scheme change.

Required background. Use anomaly coefficients and central charges, conformal geometry and maps, UV divergences and the area law, and mutual information. Helpful background. Defect entropy and monotonicity and entropy counterterms clarify defect data and scheme freedom.

Local expansion and universal remainder

For a smooth entangling surface $\Sigma$ of characteristic size $L$ in $d$ spacetime dimensions, a regulated vacuum entropy has the schematic expansion

$$S_\Sigma(\epsilon) =\sum_{k=0}^{d-2} \frac{c_k[\Sigma]}{\epsilon^{d-2-k}} +s_{\log}[\Sigma]\log\frac{L}{\epsilon} +S_{\rm finite}[\Sigma]+o(1).$$

The power divergences are integrals of local geometric densities and depend on the regulator. In even-dimensional CFTs, the logarithmic coefficient is often fixed by Weyl-anomaly data for a specified smooth shape. In odd dimensions, the sphere’s finite constant is a distinguished universal quantity after the conventional subtraction. Neither statement makes an arbitrary constant for an arbitrary region universal.

The chapter diagram separates geometric universal terms from correlation, mixed-state, and multipartite constructions.

Universal geometric coefficients arise after local ultraviolet terms are identified and subtracted. Negativity, reflected entropy, mutual information, and entropy-cone statements answer different questions and require additional constructions. Schematic.

Sphere and interval checks

In a two-dimensional CFT vacuum, one interval of length $\ell$ has

$$S(\ell)=\frac{c}{3}\log\frac{\ell}{\epsilon}+c_1,$$

where the logarithmic coefficient is universal and $c_1$ is regulator dependent. In higher-dimensional CFTs, mapping a ball’s causal development to a thermal state on hyperbolic space relates the sphere term to anomaly or sphere-free-energy data; the state and map are essential, as shown by Casini, Huerta, and Myers 2011, §§2–3.

A robust regulator comparison computes the same sphere or interval with two cutoff schemes, fits all allowed local terms, and tests whether the proposed logarithmic coefficient or subtracted sphere constant agrees. Adding a finite local counterterm is an explicit test of scheme invariance.

What can change

The validity figure distinguishes potential universal remnants from choices that must remain visible.

Anomaly-controlled logarithms and carefully subtracted sphere or topological terms can be universal. Local counterterms, corners, boundaries, state preparation, and algebra choices can shift other finite pieces. Schematic.

Moving the cutoff surface with the entangling surface, changing from a smooth sphere to a cornered lattice region, or comparing distinct state preparations can change the fitted constant. Universality is therefore a statement about a precisely matched family, not a label attached to the last term of an expansion.

References

• 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.

Further reading

• Calabrese, Pasquale, and John Cardy. “Entanglement Entropy and Quantum Field Theory.” Journal of Statistical Mechanics (2004): P06002. DOI. Open preprint.