Sphere Entanglement and Three-Dimensional F Monotonicity

In a 2+1-dimensional relativistic vacuum, the entropy of a disk contains a perimeter divergence and a finite fixed-point constant. The differential combination $\mathcal F(R)=(R\partial_R-1)S(R)$ removes the perimeter term, equals the sphere free energy at a CFT, and decreases along the standard unitary Lorentz-invariant flow. Raw disk entropy is neither finite nor the $F$ quantity.

Required background. Information measures along RG flows fixes the scale family and endpoint criteria.

Helpful background. Monotonicity and flow constraints supplies the fixed-point theorem context; universal terms and geometry explains the disk constant.

The disk F-function

For a smooth disk of radius $R$ with cutoff $\epsilon$,

$$S(R)=\alpha\frac{R}{\epsilon}-F+o(1)$$

at a conformal fixed point. Therefore

$$\mathcal F(R) =\left(R\frac{d}{dR}-1\right)S(R) =F.$$

Away from a fixed point, $\mathcal F$ is a dimensionless function of ratios such as $R/\xi$. It is not obtained by simply discarding the largest fitted term: the derivative specifies the subtraction and its numerical error.

The scale diagram identifies $F$ as a fixed-point anchor reached through a controlled regional family.

The disk function belongs to the dimension-specific theorem branch. The perimeter subtraction, disk family, and continuum limit must be held fixed before $\mathcal F(R)$ is compared with UV and IR sphere free energies. Schematic and not to scale.

Why it decreases

The entropic proof applies strong subadditivity to many boosted disks whose boundaries lie on a common null cone. Lorentz invariance controls the geometry, while the Markov property of the CFT vacuum removes otherwise dangerous “wiggly-surface” contributions in the continuum limit. The resulting concavity inequality is

$$S''(R)\le0,$$

so

$$\mathcal F'(R)=R S''(R)\le0.$$

For a flow between conformal fixed points,

$$F_{\rm UV}\ge F_{\rm IR}.$$

The null-cone strong-subadditivity construction and concavity inequality are derived in Casini and Huerta 2012, §§ II–III.

This is the entropic realization of the three-dimensional $F$ theorem. The theorem assumes a relativistic vacuum and the usual unitarity properties. Boundary RG flows, defects, finite density, or nonunitary theories require modified statements.

Free massive example

For a free massive field, write $x=mR$. The UV limit $x\ll1$ approaches the massless CFT value, while the gapped IR tends to $F_{\rm IR}=0$ after local terms have been removed. The crossover $\mathcal F(x)$ is a clean benchmark because the covariance-matrix entropy is computable on a radial or spatial lattice.

A numerical extraction should not differentiate noisy values once and stop. Fit the entropy over a local $R$ window, propagate the covariance of the fit into $RS'-S$, and repeat at several $R/\epsilon$. The checks are:

$$\mathcal F(0)=F_{\rm CFT},\qquad \mathcal F(\infty)=0,\qquad \mathcal F'(R)\le0$$

within controlled discretization and finite-volume errors. A narrow window can appear flat even when it lies between the two scaling regimes.

Fixed-point F versus finite sphere free energy

At a three-dimensional CFT, $F=-\log |Z_{S^3}|$ in the standard unitary convention and equals the universal disk constant. Away from criticality, the finite part of a sphere partition function can be shifted by local terms, and it need not equal the disk differential function point by point. Endpoint equality does not make all interpolating prescriptions identical.

The sphere-free-energy formulation and supersymmetric fixed-point tests appear in Jafferis et al. 2011, §§ 2–4, while the differential entropy prescription is analyzed in Liu and Mezei 2013, §§ 2.1 and 4.

The independent information is therefore: a theorem about the endpoint ordering, a particular entropic interpolant with its geometric proof, and other scheme-dependent finite functions that can be useful without being the theorem.

Validity map

Validity map for $F$ monotonicity. The disk family, regulator, relativistic vacuum, and theorem geometry are essential. A decreasing subtraction in a nonunitary model or a boundary flow is a diagnostic, not this theorem. Schematic and not to scale.

Common pitfalls

Identifying the raw constant term by a single fit. The perimeter coefficient and finite constant are correlated at finite cutoff. Use the differential combination and refine the cutoff.

Treating endpoint equality as scheme equality. Different finite prescriptions can agree at CFTs and disagree in the crossover. Compare their allowed local terms before comparing curves.

References

• Casini, Horacio, and Marina Huerta. “On the RG Running of the Entanglement Entropy of a Circle.” Physical Review D 85 (2012): 125016. DOI.
• Jafferis, Daniel L., Igor R. Klebanov, Silviu S. Pufu, and Benjamin R. Safdi. “Towards the F-Theorem: $\mathcal N=2$ Field Theories on the Three-Sphere.” Journal of High Energy Physics 2011, no. 6 (2011): 102. DOI.
• Liu, Hong, and Mark Mezei. “A Refinement of Entanglement Entropy and the Number of Degrees of Freedom.” Journal of High Energy Physics 2013, no. 4 (2013): 162. DOI.