Relevant Deformations and Entanglement Crossover Functions

A relevant deformation replaces exact scale invariance by a finite crossover. The fixed-point scaling dimension determines the leading crossover exponent, but a universal curve emerges only after the observable, scaling field, regulator, volume, and correction terms have been fixed. Endpoint powers and logarithms are usually more robust than an unconstrained interpolation.

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

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

Keep the chapter’s separation of scale roles, comparison table, and independent validity gates in view: a successful collapse is not, by itself, a monotonicity theorem.

Scaling fields and crossover variables

Perturb a UV CFT by a scalar operator,

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

At the fixed point, the linearized RG eigenvalue is

$$y=d-\Delta_{\mathcal O}>0, \qquad [g]={\rm mass}^{y}.$$

This statement concerns a normalized scaling field. A bare coupling can mix with other couplings and differ from it by analytic redefinitions and a nonuniversal metric factor. On a massive branch the scaling field generates a correlation length

$$\xi=A_\xi\lvert g\rvert^{-1/y} \left[1+O\!\left( \left(\frac{\lvert g\rvert}{\Lambda^y}\right)^{\omega/y} \right)\right],$$

where $A_\xi$ depends on the normalization of $g$, $\Lambda$ is a fixed microscopic reference scale, and $\omega>0$ is the leading irrelevant exponent. On a flow to another CFT, the same expression defines a crossover scale, not a finite physical correlation length. A convenient dimensionless variable is

$$x=gR^y, \qquad \lvert x\rvert =A_\xi^y(R/\xi)^y \left[1+O\!\left( \left(\frac{\lvert g\rvert}{\Lambda^y}\right)^{\omega/y} \right)\right].$$

Thus $x$ and $R/\xi$ are equivalent only to leading scaling order; the correction factor must be retained in a precision crossover fit.

Two standard mass deformations illustrate why $m$ and $m^2$ cannot be interchanged as a matter of notation alone.

Theory and perturbation	Operator dimension	Scaling field	Eigenvalue	Region variable
Free scalar, $\frac12m^2\phi^2$	$d-2$ at the Gaussian fixed point	$g=m^2/2$	$y=2$	$(mR)^2$
Free Dirac field, $m\bar\psi\psi$	$d-1$	$g=m$	$y=1$	$mR$

The physical crossover may be plotted against $mR$ in both examples, but its relation to the Lagrangian scaling field is different.

Corrections that define the scaling window

For a dimensionless, renormalized information observable $\mathcal I$, a finite-regulator ansatz is

$$\begin{aligned} \mathcal I(R,g,a,L) ={}&\Phi(x) +A(x)(a/R)^p +B(x)(R/L)^q \\ &+u\,(a/R)^\omega C(x)+\cdots . \end{aligned}$$

Here $a$ is the short-distance cutoff, $L$ the outer size, and $u$ a dimensionless amplitude of the leading irrelevant lattice field. The exponents $p,q,\omega$ and amplitudes are properties of the regulator, boundary conditions, and observable; they are not optional fit decorations. A scaling window is established only when the inferred $\Phi$ is stable as $a/R$ and $R/L$ are reduced independently.

At small $x$, finite-region conformal perturbation theory can give

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

but symmetries or vanishing one-point functions can remove coefficients. Coincident insertions and resonances can instead produce $x^n\log\lvert x\rvert$ terms. The coefficients of analytic powers are not automatically universal: contact prescriptions and the normalization of $g$ matter. Replica perturbation theory and its contact terms are developed in Rosenhaus and Smolkin 2014, §§2–4.

In a gapped phase, connected correlations have exponential long-distance tails. Entanglement observables can additionally contain boundary-local expansions in curvature and inverse powers of the gap. Which local terms survive depends on the renormalization prescription. A flow to another CFT instead approaches a nonzero IR anchor with corrections controlled by irrelevant operators of the IR theory.

Exact free-field anchor

For one interval of length $R$ in the vacuum of the massive scalar in $1+1$ dimensions, define

$$\mathfrak c(t)=R\frac{dS}{dR}, \qquad t=mR, \qquad c_E(t)=3\mathfrak c(t).$$

The cutoff-independent function obeys $c_E(0)=1$, the central charge of the noncompact free boson, with a logarithmically delicate approach caused by its zero mode. At large $t$, Casini and Huerta obtain

$$\mathfrak c(t)\sim\frac14tK_1(2t), \qquad c_E(t)\sim\frac34tK_1(2t),$$

where $K_1$ is a modified Bessel function Casini and Huerta 2005, Eq. (88). Thus the IR tail is not an arbitrary exponential fit:

$$c_E(t)\sim \frac{3\sqrt\pi}{8}\sqrt t\,e^{-2t} \left(1+\frac{3}{16t}+O(t^{-2})\right).$$

A lattice benchmark should therefore compare the continuum-extrapolated interval function with both the UV anchor and this IR asymptotic, while treating the periodic scalar zero mode separately. Dirichlet boundaries, removal of the spatial zero mode, or a controlled $mL>0$ extrapolation are acceptable choices if stated explicitly.

A reproducible scaling-window stress test

The following test is deliberately synthetic: it checks the extraction protocol and does not claim to be free-scalar data. Generate all combinations

$$R\in\{1,2,4,8\},\quad g\in\{0.0025,0.01,0.04,0.16\},$$ $$R/a\in\{4,8,16,32,64\}, \qquad L/R\in\{4,8,16\},$$

in arbitrary but fixed units, and assign

$$\mathcal I_n= \frac{1}{1+gR^2} +0.8(a/R)^2 +0.5(R/L)^2 +2\times10^{-4}\sin n,$$

where $n=1,\ldots,240$ is the lexicographic row number. Fit the intentionally incomplete two-parameter model

$$\mathcal I_{\rm fit}=\frac{1}{1+c\,gR^{y}}$$

on the grid $\Delta y=\Delta c=0.002$, with $1.4\le y\le2.6$ and $0.8\le c\le1.2$. The deterministic result is:

Accepted rows	Number of rows	Best $y$	Best $c$	RMS residual
$R/a\ge32$, $L/R\ge16$	32	2.002	0.984	$1.487\times10^{-3}$
$R/a\ge16$, $L/R\ge8$	96	2.008	0.954	$4.903\times10^{-3}$
All rows, including $R/a=4$ and $L/R=4$	240	2.030	0.822	$2.790\times10^{-2}$

The control rule is fixed before looking at the broad fit: accept a scaling exponent only if $\lvert y-2\rvert\le0.01$ and the RMS residual is below $5\times10^{-3}$. The clean and nominal windows pass; the all-row fit fails both the exponent and residual tests. The $0.028$ exponent displacement is fourteen grid spacings, much larger than the quoted numerical resolution. The strongest surviving statement is therefore: the controlled window reproduces the imposed Gaussian eigenvalue $y=2$; data outside that window do not support the same collapse.

For physical data, replace the synthetic curve by a covariance-matrix calculation, propagate the independently measured uncertainty in $\xi$ as a horizontal uncertainty, and repeat the cut scan. The chapter’s common free-scalar dataset is the natural numerical continuation of this test.

Interpretation and limitations

A crossover curve is not automatically monotone even if its endpoint central charges satisfy a theorem. Thermal length, defect distance, boundary conditions, and additional relevant couplings can introduce independent variables. A visually good one-variable collapse can also be manufactured by rescaling every curve separately; predictive collapse uses one scaling field and one jointly fitted correction model.

The uncertainty report should include at least: the range of $a/R$ and $R/L$; zero-mode and boundary prescription; covariance or entropy precision; correlation-length estimator and its covariance with the horizontal axis; fit form; leave-one-resolution-out stability; and the change when the shortest and longest scales are removed.

Exercises

1. Scalar and Dirac scaling variables

At a Gaussian fixed point in $d$ spacetime dimensions, show that the scalar mass-squared deformation and the Dirac mass deformation have eigenvalues $2$ and $1$, respectively. Express $\xi$ in terms of the corresponding Lagrangian coupling.

Solution

The free scalar has $[\phi]=(d-2)/2$, hence $[\phi^2]=d-2$. The coefficient of $\frac12\phi^2$ therefore has dimension $d-(d-2)=2$. With $g_s=m^2/2$, the linearized eigenvalue is $y_s=2$ and

$$\xi=A_s\lvert g_s\rvert^{-1/2} =A_s\sqrt2/m.$$

The factor $A_s\sqrt2$ is nonuniversal; physically one normally chooses the mass normalization for which $\xi=1/m$.

For a free Dirac field, $[\psi]=(d-1)/2$, so $[\bar\psi\psi]=d-1$. Its coefficient $g_f=m$ has dimension one, hence $y_f=1$ and $\xi=A_f/\lvert m\rvert$. In both cases the physical regional variable can be written $mR$, but $g_sR^{y_s}\propto(mR)^2$ whereas $g_fR^{y_f}=mR$.

2. The massive-scalar infrared tail

Starting from $c_E(t)\sim\frac34tK_1(2t)$, derive its first large-$t$ asymptotic term and explain why a fit to $Ae^{-t}$ has the wrong physical decay rate.

Solution

For fixed order $\nu$,

$$K_\nu(z)\sim \sqrt{\frac{\pi}{2z}}e^{-z} \left[1+\frac{4\nu^2-1}{8z}+O(z^{-2})\right].$$

Set $\nu=1$ and $z=2t$:

$$K_1(2t)\sim \sqrt{\frac{\pi}{4t}}e^{-2t} \left(1+\frac{3}{16t}+O(t^{-2})\right).$$

Multiplication by $3t/4$ gives

$$c_E(t)\sim \frac{3\sqrt\pi}{8}\sqrt t\,e^{-2t} \left(1+\frac{3}{16t}+O(t^{-2})\right).$$

The exponent is $2mR$, reflecting the leading two-particle correlation contribution. A fit to $Ae^{-mR}$ therefore has both the wrong exponential rate and the wrong $\sqrt{mR}$ prefactor.

3. Diagnose the failed synthetic collapse

For the synthetic dataset, compute the largest deterministic correction from the two coarsest controls, $R/a=4$ and $L/R=4$, ignoring the small sine term. Compare it with the corresponding correction at $R/a=32$ and $L/R=16$. Why is an exponent drift expected when corrections are omitted from the fit?

Solution

At the coarse controls,

$$0.8(a/R)^2+0.5(R/L)^2 =\frac{0.8}{16}+\frac{0.5}{16} =0.08125.$$

At the clean controls,

$$\frac{0.8}{32^2}+\frac{0.5}{16^2} =0.00078125+0.001953125 =0.002734375.$$

The coarse offset is almost thirty times larger. It is also not a function of $gR^2$ alone, so the incomplete fit cannot absorb it with one universal curve. It compensates by moving both the horizontal exponent $y$ and the curve normalization $c$, producing the observed $y=2.030$ and $c=0.822$. The failed residual and exponent controls correctly prevent that compensation from being called universal scaling.

References

• Casini, Horacio, and Marina Huerta. “Entanglement and Alpha Entropies for a Massive Scalar Field in Two Dimensions.” Journal of Statistical Mechanics: Theory and Experiment 2005, no. 12 (2005): P12012. DOI; Open preprint.
• Rosenhaus, Vladimir, and Michael Smolkin. “Entanglement Entropy: A Perturbative Calculation.” Journal of High Energy Physics 2014, no. 12 (2014): 179. DOI; Open preprint.

Further reading

• Cardy, John. Scaling and Renormalization in Statistical Physics. Cambridge: Cambridge University Press, 1996. DOI.