Entropic Monotones in Two Dimensions

For the vacuum of a unitary Lorentz-invariant QFT in 1+1 dimensions, interval entropy defines a genuine entropic $c$-function. Strong subadditivity applied to boosted intervals gives a differential inequality, and conformal scaling fixes its UV and IR normalizations. The theorem is geometric and dimension-specific; it does not automatically extend to thermal, nonrelativistic, or nonunitary systems.

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

Helpful background. Monotonicity and flow constraints supplies the CFT endpoint meaning; strong subadditivity supplies the entropy inequality.

The interval c-function

Let $S(R)$ be the vacuum entropy of an interval whose proper length is $R$. Define

$$c_E(R)=3R\frac{dS(R)}{dR}.$$

At a CFT,

$$S(R)=\frac{c}{3}\log\frac{R}{\epsilon}+s_0,$$

so $c_E(R)=c$. The derivative removes the additive cutoff-dependent constant. Along a massive flow, $c_E$ becomes a function of $mR$ and tends to the UV and IR central charges when those endpoints are conformal.

The structure diagram places this theorem on the fixed-point branch rather than the channel branch.

The two-dimensional entropic $c$-function occupies the dimension-specific theorem branch: the interval family and continuum derivative must be fixed before its endpoints are compared. Schematic and not to scale.

Strong subadditivity and Lorentz geometry

Choose boosted intervals $B$ and $C$ whose causal diamonds have intersection $A$ and causal completion of their union $D$. Vacuum Lorentz invariance makes the entropy depend only on proper length. Strong subadditivity gives

$$S(B)+S(C)\ge S(A)+S(D).$$

Taking a symmetric infinitesimal configuration yields

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

and therefore

$$\frac{dc_E}{dR} =3\bigl[S'(R)+RS''(R)\bigr]\le0.$$

This boosted-diamond derivation and its normalization are given explicitly in Casini and Huerta 2007, pp. 7032–7035.

The normalization factor $3$ is chosen so that the fixed-point value is the Virasoro central charge in the standard interval convention. The proof uses causal diamonds and Lorentz boosts; ordinary nested equal-time intervals alone do not supply the same differential inequality.

Hypotheses and endpoints

The precise conclusion is

$$c_{\rm UV}=\lim_{R\to0}c_E(R) \ge \lim_{R\to\infty}c_E(R)=c_{\rm IR},$$

when the limits reach conformal fixed points and the vacuum assumptions hold. A gapped IR has $c_{\rm IR}=0$ after correlations saturate. Equality throughout signals scale-independent interval entropy derivative, as at a fixed point.

Finite temperature introduces the thermal length $\beta$ and changes the large-$R$ entropy to an extensive thermal term. A Lifshitz theory lacks the Lorentz construction. Nonunitary theories can violate the positivity properties behind standard entropy inequalities or have an effective central charge distinct from $c$. Each case needs a replacement statement.

Free massive fields

For a free massive Dirac field, $c_E(mR)$ decreases smoothly from $1$ to $0$. A real scalar approaches $c=1$ in the UV but its zero-mode behavior makes the small-mass limit numerically delicate. In both cases a reliable computation diagonalizes a regulated covariance matrix, differentiates only after cutoff convergence, and compares several values of $R/\epsilon$ at fixed $mR$.

The free-field integral equations, lattice checks, and scalar zero-mode discussion are collected in Casini and Huerta 2009, §§ 3.1–3.3.

The test is not merely that the curve decreases. It should reproduce the CFT normalization at $mR\ll1$, decay in the massive regime, and remain stable under lattice refinement. A derivative stencil can manufacture a local upward fluctuation if the entropy errors are correlated or the grid is too coarse.

Validity boundary

The relevant theorem gate is the third one in the map: a common interval family and regulator are necessary but not sufficient; Lorentz invariance, the vacuum, unitarity, and 1+1-dimensional geometry do the remaining work.

Validity map for the entropic $c$-theorem. If the state is thermal, the dynamics is non-Lorentz-invariant, or the regional prescription changes with $R$, one may still plot a regulated function, but the theorem no longer licenses monotonicity. Schematic and not to scale.

Common pitfalls

Dropping the factor of three. $RS'(R)$ is monotone too, but it equals $c/3$ at a CFT. State the normalization before comparing endpoints.

Using a finite-temperature interval as the vacuum theorem. The thermal entropy grows extensively at large $R$. The inverse temperature is an additional scale and changes the conclusion.

Exercises

At a CFT, insert $S(R)=(c/3)\log(R/\epsilon)+s_0$ into the definition of $c_E$ and verify both fixed-point normalization and saturation of the differential inequality.

Solution

Differentiation gives $S'=c/(3R)$ and $S''=-c/(3R^2)$. Hence $c_E=3R S'=c$ and $RS''+S'=0$.

References

• Casini, Horacio, and Marina Huerta. “A c-Theorem for the Entanglement Entropy.” Journal of Physics A: Mathematical and Theoretical 40 (2007): 7031–7036. DOI.
• Casini, Horacio, and Marina Huerta. “Entanglement Entropy in Free Quantum Field Theory.” Journal of Physics A: Mathematical and Theoretical 42 (2009): 504007. DOI.