Relevant Deformations and Entanglement Crossover Functions

A relevant deformation replaces fixed-point scale invariance by a crossover governed by dimensionless combinations of the coupling and region size. Universal scaling functions can be extracted only in a window where the cutoff, finite volume, and irrelevant operators are all subleading. Their endpoint asymptotics are often more robust than the full interpolating curve.

Required background. Modes, regions, and factorization scales supplies the multiscale hierarchy; universal scaling functions supplies data collapse; renormalized entropy schemes fixes the finite observable.

Helpful background. Monotonicity and flow constraints identifies endpoint constraints; fidelity susceptibility under deformations supplies a local response diagnostic.

Scaling variables from the deformation

Perturb a UV CFT by

$$I=I_{\rm CFT}+g\int d^d x\,\mathcal O(x), \qquad \Delta<d.$$

The coupling has RG eigenvalue $y=d-\Delta>0$. Ignoring operator mixing for the moment, it generates

$$\xi\sim |g|^{-1/y}, \qquad x=gR^y\sim \operatorname{sgn}(g)(R/\xi)^y.$$

A dimensionless information observable then has the scaling form

$$\mathcal I(R,g,\epsilon,L_{\rm box}) =\Phi(x) +\left(\frac{\epsilon}{R}\right)^p A(x) +\left(\frac{R}{L_{\rm box}}\right)^q B(x) +uR^{-\omega}C(x)+\cdots.$$

The last three terms are cutoff, finite-volume, and irrelevant-operator corrections. A successful collapse must show that they decrease when the corresponding controls are refined.

The scaling-field construction and correction-to-scaling logic are developed in Cardy 1996, ch. 3, §§ 3.2–3.4.

Relevant deformations occupy the finite-crossover branch. The universal variable $gR^{d-\Delta}$ is meaningful only after the cutoff ratio, box size, region family, and renormalized information prescription are controlled. Schematic and not to scale.

Conformal perturbation and contact terms

At small $x$, one may expand

$$\Phi(x)=\Phi(0)+b_1x+b_2x^2+\cdots,$$

but the first nonzero power depends on symmetries, one-point functions, and the subtraction scheme. Integrated correlators can have contact divergences; relevant or marginal resonances can produce terms such as $x^n\log x$. The coefficients are not universal merely because the power of $x$ is fixed by dimensional analysis.

At large $R/\xi$, a gapped phase yields boundary-local contributions plus exponentially small correlations, whereas a flow to another CFT approaches a nonzero IR anchor with corrections from its leading irrelevant operator. Fit the appropriate asymptotic form rather than a generic polynomial.

The replica perturbation and contact-term analysis is given in Rosenhaus and Smolkin 2014, §§ 2–4; its Ward-identity form and entropy-flow equation appear in Rosenhaus and Smolkin 2014, Eqs. (6)–(12).

Data-collapse workflow

Suppose a lattice calculation provides $S(R;g,a,L_{\rm box})$. A defensible extraction proceeds as follows:

1. construct the renormalized observable using one fixed prescription;
2. determine the physical correlation length $\xi(g,a)$ independently;
3. plot against $R/\xi$, retaining only $R/a$ and $L_{\rm box}/R$ values that pass refinement checks;
4. fit all datasets jointly with explicit cutoff and finite-size corrections; and
5. repeat after dropping the shortest and longest scales.

The uncertainty on $\xi$ is correlated with the horizontal axis and must enter the fit. A visually good collapse can be produced by rescaling each curve independently; that procedure has no predictive content.

Free massive example

For a free field, $g=m$ or $m^2$ depending on the operator convention, but the physical variable is $mR$. Interval and disk functions approach their CFT constants at $mR\ll1$ and their gapped values at $mR\gg1$. The crossover around $mR\sim1$ is an ideal regulator benchmark.

Compare two mass values with the same $mR$ but different $R/a$. Agreement tests scaling. Compare two system sizes with the same $mR$ and $R/a$ to test finite volume. Finally, fit the UV and IR tails independently. Only then should a common $\Phi(mR)$ be quoted.

Claim boundaries

A crossover function is not automatically monotone, even when its endpoints obey a theorem. A finite region can also probe defect, boundary, or thermal scales not included in $x$. If several relevant couplings are present, theory space is multidimensional and one-variable collapse can fail for physical reasons.

Validity map for crossover functions. Passing continuum and fixed-family gates licenses a scaling comparison; monotonicity still requires a theorem, and information loss still requires a channel. Schematic and not to scale.

Common pitfalls

Fitting before establishing a scaling window. Short distances retain cutoff artifacts and long distances retain finite-size effects. Remove both before claiming universality.

Treating correction exponents as optional decoration. Irrelevant operators can bend a curve enough to imitate nonmonotonicity or a false plateau. Include them or bound their effect.

References

• Cardy, John. Scaling and Renormalization in Statistical Physics. Cambridge: Cambridge University Press, 1996. DOI.
• Rosenhaus, Vladimir, and Michael Smolkin. “Entanglement Entropy: A Perturbative Calculation.” Journal of High Energy Physics 2014, no. 12 (2014): 179. DOI.
• Rosenhaus, Vladimir, and Michael Smolkin. “Entanglement Entropy Flow and the Ward Identity.” Physical Review Letters 113 (2014): 261602. DOI.