Separating Cutoff, Region, Correlation, and RG Scales

An information observable in QFT generally depends on several lengths at once. The ultraviolet cutoff $\epsilon$, region size $R$, inter-region separation $L$, correlation length $\xi$, deformation scale, and renormalization length $\mu^{-1}$ play distinct roles, but they need not be independent: in a free massive continuum theory, for example, $\xi=m^{-1}$. A controlled flow claim records coincidences as carefully as differences, identifies which dimensionless ratio is varied, and states the order of regulator, volume, and endpoint limits.

Required background. Regulated subregion entropy supplies the cutoff-dependent regional entropy; relative entropy in QFT supplies regulator-independent comparisons when a common algebra exists; UV and IR fixed points supplies the endpoint language.

Helpful background. Lattice-to-continuum entropy explains cutoff refinement in a concrete regulator.

The chapter overview first separates the scale roles, then compares candidate diagnostics in its comparison table and applies independent validity gates. This page supplies the calculation behind those checks.

Six roles that may coincide or vanish

Near a UV fixed point, let $\mathcal O$ be a scaling operator of dimension $\Delta$ and let $g_R(\mu)$ be its renormalized coupling. With $y=d-\Delta>0$, a regional quantity can have the schematic form

$$I_A=I_A\!\left( \frac{R}{\epsilon},\frac{L}{R},\frac{R}{\xi}, g_R(\mu)R^y,\mu R;\text{shape,state,prescription} \right).$$

The formula is local to the scaling regime. Away from it, scaling fields can mix and the full running couplings replace a single monomial. Even near the fixed point,

$$\xi=A_\pm |g_R|^{-1/y} \left[ 1+B_\pm\left(\frac{|g_R|}{\Lambda^y}\right)^{\omega/y} +\cdots \right]$$

contains phase-dependent amplitudes $A_\pm,B_\pm$, a fixed microscopic reference scale $\Lambda$, and corrections governed by an irrelevant exponent $\omega>0$. The ratio inside the correction is dimensionless. Thus $R/\xi$ and $g_RR^y$ are related rather than freely adjustable in a one-coupling scaling theory.

Role	Operational definition	What happens in the benchmark
Cutoff $\epsilon$	shortest resolved length	$\epsilon=a$
Region size $R$	length of each selected interval	$R=N_Aa=1$
Separation $L$	gap between two selected intervals	$L=N_La=R$; it is absent for a single interval
Correlation length $\xi$	inverse exponential decay rate of equal-time correlations	$\xi_{\rm lat}=a/[2\operatorname{arsinh}(ma/2)]$
Deformation length	length built from the relevant coupling	$m^{-1}$; it approaches $\xi_{\rm lat}$ as $a\to0$
Renormalization length $\mu^{-1}$	reference used for renormalized couplings and operators	$\mu=R^{-1}=1$; the free mass does not run

For an interacting observable, residual $\mu$ dependence is removed only after the running couplings, operator renormalization, and allowed counterterms are combined. In the free-chain benchmark, $\mu$ is merely a declared reporting convention.

Order of continuum and endpoint limits

The safe order depends on the question, but it must be declared. To extract a continuum crossover from a lattice model, first hold $R/\xi$, $L/R$, and $L_{\rm box}/R$ fixed while taking

$$\frac{R}{\epsilon}\longrightarrow\infty,$$

then study $R/\xi\to0$ or $R/\xi\to\infty$. Reversing these steps can erase the scaling window. Taking $mR\to0$ at fixed $R/\epsilon$, for example, approaches a nearly critical finite lattice rather than the continuum CFT.

The hierarchy needed for a massive interval calculation is

$$\epsilon\ll R\ll \xi \quad\text{(UV scaling)}, \qquad \epsilon\ll \xi\ll R \quad\text{(IR saturation)}.$$

Between them, $R\sim\xi$ is the crossover. A numerical analysis should display $R/\epsilon$, $mR$, $L/R$, and $L_{\rm box}/R$ separately. Agreement at one lattice spacing is not a continuum result.

Spatial enlargement is not coarse graining

For $\mathcal A(A)\subset\mathcal A(B)$, restriction of normal states from $\mathcal A(B)$ to $\mathcal A(A)$ contracts Araki relative entropy: the relative-modular definition and its finite-dimensional reduction are Araki 1976, § 1, Eqs. (1.1)–(1.2), pp. 809–810, while contraction under the unital inclusion is Uhlmann 1977, Proposition 18, p. 31. In a type-I lattice regulator this is the familiar state channel dual to the algebra inclusion; a continuum local algebra need not admit a density matrix or tensor factorization. Enlarging a nested region, by contrast, gives access to more observables. Wilsonian RG integrates or reorganizes high-momentum fluctuations and changes an effective action. These operations can be related by an additional construction, but they are not identical.

This distinction controls monotonicity directions. Relative entropy decreases under restriction, yet a region-size derivative can have either sign before a theorem uses Lorentz symmetry, strong subadditivity, and a particular geometric family. Similarly, a running coupling $g_R(\mu)$ changes with $\mu$ while a physical observable is $\mu$-independent only after renormalization is completed.

Periodic harmonic-chain benchmark

The reproducible application uses the ground state of a real massive harmonic chain with $N_{\rm box}$ sites and periodic boundary conditions,

$$H=\frac12\sum_{n=0}^{N_{\rm box}-1} \left[p_n^2+m^2q_n^2+\frac{(q_{n+1}-q_n)^2}{a^2}\right], \qquad [q_n,p_j]=i\delta_{nj}, \qquad q_{N_{\rm box}}=q_0.$$

Its normal-mode frequencies and ground-state correlators are

$$\omega_k^2=m^2+\frac{4}{a^2}\sin^2\!\frac{\pi k}{N_{\rm box}},$$ $$X_r=\frac{1}{2N_{\rm box}}\sum_k \frac{\cos(2\pi kr/N_{\rm box})}{\omega_k}, \qquad P_r=\frac{1}{2N_{\rm box}}\sum_k \omega_k\cos\!\frac{2\pi kr}{N_{\rm box}}.$$

For a set $V$ of sites, let $\nu_j$ be the positive square roots of the eigenvalues of $X_VP_V$. The entropy in nats is

$$S(V)=\sum_j\left[ \left(\nu_j+\frac12\right)\log\left(\nu_j+\frac12\right) -\left(\nu_j-\frac12\right)\log\left(\nu_j-\frac12\right) \right].$$

The calculation fixes $R=1$, $L/R=1$, and uses

$$t=mR\in\{0.25,0.35,0.5,0.75,1,1.5,2,3,4,5,6\},$$ $$N_A=R/a\in\{16,24,32,48\}, \qquad L_{\rm box}/R\in\{12,16\}.$$

At every fixed $t$, changing $N_A$ changes $ma=t/N_A$ while preserving the physical crossover point. The entropic function is estimated within the same chain, at fixed $a$, $m$, and $N_{\rm box}$, by

$$c_E^{(h)}(t,a)= 3N_A\frac{S(N_A+h)-S(N_A-h)}{2h}, \qquad h=1,2.$$

The central $h=2$ result is fitted linearly in $(a/R)^2$. The quoted deterministic resolution-sensitivity envelope is defined as the sum of three observed changes: all four cutoff points versus the finest three, $L_{\rm box}/R=16$ versus $12$, and $h=2$ versus $h=1$. It is an operational convergence diagnostic, not a statistical confidence interval or a rigorous bound on every omitted effect.

$t=mR$	Continuum $c_E$	Sensitivity envelope	Dominant check
0.25	0.420732	0.009573	finite volume and the scalar zero mode
0.50	0.267819	0.000753	finite volume
1.00	0.112797	0.000011	stencil and box checks comparable
2.00	0.0190698	0.0000089	derivative extraction
4.00	0.00046545	0.00000251	derivative extraction
6.00	0.00001012	0.00000030	derivative extraction

The decrease is consistent with the two-dimensional entropic $c$-theorem, but these eleven points do not prove the theorem. The relatively large $t=0.25$ box error is expected for the noncompact scalar zero mode; it is precisely why a volume diagnostic belongs beside cutoff refinement. The continuum interval normalization, massive saturation, covariance method, and zero-mode qualification are developed in Calabrese and Cardy 2004, §§ III.1, IV.1–IV.2, and V and Casini and Huerta 2009, §§ 2.2.1, 3.1.1, 3.1.4, and 3.1.8.

The complete machine-readable calculation is available as JSON and CSV.

Adversarial test: fixed ma is not a continuum limit

Now deliberately hold $ma=0.04$ fixed while increasing $N_A=R/a$. Because

$$mR=(ma)N_A,$$

the physical crossover variable changes at the same time as the nominal resolution.

$R/a=N_A$	$ma$	$mR$	$c_E^{(2)}$
12	0.04	0.48	0.283014
20	0.04	0.80	0.161381
32	0.04	1.28	0.0696258
48	0.04	1.92	0.0221862
64	0.04	2.56	0.00690145

If the horizontal axis were mislabeled merely as “finer lattice,” this table would manufacture an impressive monotone continuum flow. The downgrade is exact: it is a regulated sequence along which both $a/R$ and $mR$ vary. The strongest surviving claim is only that the regulated harmonic-chain observable decreases along this mixed path. A continuum crossover claim requires comparison at fixed $mR$, fixed $L/R$, and controlled $L_{\rm box}/R$.

Common pitfalls

Hiding the cutoff in a dimensionless mass. Holding $ma$ fixed while increasing the number of sites does not hold $mR$ fixed. Report both $mR$ and $R/a$.

Calling every $R$ dependence RG flow. Region enlargement changes the accessible algebra. An RG interpretation requires an additional theorem, scaling argument, or channel construction.

Taking endpoint limits before the continuum limit. A lattice plateau can be produced by finite size or by $R$ approaching only a few cutoff units. Repeat the extraction at fixed physical ratios under cutoff refinement.

Exercises

1. Derive the lattice correlation length

Show that the zero-frequency pole of the lattice dispersion gives

$$\xi_{\rm lat}=\frac{a}{2\operatorname{arsinh}(ma/2)},$$

and find its continuum expansion.

Solution

Continue the spatial momentum to $k=i\kappa$. The pole condition is

$$0=m^2-\frac{4}{a^2}\sinh^2\!\frac{\kappa a}{2},$$

so $\kappa=2a^{-1}\operatorname{arsinh}(ma/2)$ and $\xi_{\rm lat}=\kappa^{-1}$. Since $\operatorname{arsinh}z=z-z^3/6+O(z^5)$,

$$\xi_{\rm lat} =\frac1m\left[1+\frac{(ma)^2}{24}+O((ma)^4)\right].$$

Thus $\xi$ and $m^{-1}$ are distinct at finite cutoff but coincide in the continuum limit.

2. Check the derivative estimator

Prove that $c_E^{(h)}$ is dimensionless, cancels an additive endpoint divergence, and has leading relative step error $O((ha/R)^2)$ for a smooth entropy.

Solution

With $R=N_Aa$,

$$\frac{dS}{dR} =\frac{S(N_A+h)-S(N_A-h)}{2ha} +O((ha)^2S'''(R)).$$

Multiplication by $3R=3N_Aa$ gives the estimator on the page and removes the unit of inverse length. If $S\mapsto S+C(a,m)$ with $C$ independent of $R$, the two occurrences of $C$ cancel. Dividing the Taylor remainder by the leading derivative gives a relative scale $O((ha/R)^2)$ when $S$ varies on the scale $R$.

3. Diagnose the mixed-limit table

Using only the first and last rows of the fixed-$ma$ table, calculate the changes in $a/R$ and $mR$. Explain why neither a cutoff extrapolation nor a fixed-$t$ monotonicity test follows.

Solution

The nominal cutoff ratio changes from $a/R=1/12$ to $1/64$, an improvement by a factor $64/12\approx5.33$. But $mR$ simultaneously changes from $0.48$ to $2.56$, also by a factor $2.56/0.48\approx5.33$. A continuum extrapolation requires $mR$ fixed while $a/R\to0$; a crossover comparison may vary $mR$ but must separately control $a/R$. This sequence does neither, so only a mixed regulated trend survives.

References

• Araki, Huzihiro. “Relative Entropy of States of von Neumann Algebras.” Publications of the Research Institute for Mathematical Sciences 11, no. 3 (1976): 809–833. DOI.
• Calabrese, Pasquale, and John Cardy. “Entanglement Entropy and Quantum Field Theory.” Journal of Statistical Mechanics: Theory and Experiment 2004 (2004): P06002. DOI.
• Casini, Horacio, and Marina Huerta. “Entanglement Entropy in Free Quantum Field Theory.” Journal of Physics A: Mathematical and Theoretical 42 (2009): 504007. DOI.
• Uhlmann, Armin. “Relative Entropy and the Wigner–Yanase–Dyson–Lieb Concavity in an Interpolation Theory.” Communications in Mathematical Physics 54, no. 1 (1977): 21–32. DOI.