cft-cmi

Module 8 — Off criticality: gap, temperature, mass

The quadratic law is a statement about adjacent intervals at a conformal fixed point. Open a gap, heat the state, or turn on a mass, and you can watch which parts survive. Heating is a conformal change of state and the law survives it, with the cross ratio transported; a gap or a mass is not, and there the recovery error becomes linear in the conditional mutual information.

proved The gap law itself is a closed form

With a gap G between A and B, traced out, the massless Dirac two-interval entropies give exactly the adjacent-interval formula with LB → LB + LG: I(A:C|B) = (1/3) log[(a+g+b)(g+b+c) / ((g+b)(a+g+b+c))] ~ (1/3) L_A L_C / L_G². Every log LG term cancels. Your browser evaluates this formula live; at LG = 96 the lattice confirms it to 0.116% at L = 6 and to 0.055% at L = 12.

Where this comes from: the adjacent-interval CMI · the gap claim · rigor/thermal_rg_gap_summary.md

numerical Separated intervals: the recovery error decays with the same power

If the quadratic law held, −log F would go like LG−4. It does not: it decays like LG−2, so the ratio −log F / I tends to a constant near 0.09, independent of the interval length. Recovery is only quadratic in the CMI at a conformal fixed point with adjacency.

Where this comes from: claim page · numerics/lattice/petz_thermal_rg.py (output petz_thermal_rg.out, section 3)

numerical Thermal states: the vacuum law at the cylinder cross ratio, and exponential shielding

KMS states obey the vacuum two-branch law at the thermal cross ratio eta_beta = sinh(pi a/bl) sinh(pi c/bl) / [sinh(pi(a+b)/bl) sinh(pi(b+c)/bl)], bl = 2 beta, the vacuum cross ratio of the images of the four endpoints under the exponential map. For LB ≫ β the error is shielded exponentially, at rate 2π in −log F and π in the CMI, in thermal-length units (v = 2 sites per unit lattice time, so ξ = β/π sites).

Where this comes from: claim page · rigor/thermal_rg_gap_summary.md §1

numerical The Markov c-function along the massive flow

With cM(R) = 3R² IR−2(2)/4, a genuine lattice CMI and so non-negative by strong subadditivity, and cE(R) = 3R (S(R+2) − S(R−2))/4: both fall monotonically from 1 to 0 along the flow, cM ≥ cE everywhere as the entropic c-theorem requires, and cM/cE → mR in the infrared — so cM is not an independent infrared quantity, it is cE times the RG time.

Both columns overshoot at small R. The plotted cM(2) reaches 1.1079 at m = 0.05, R = 6, and stays above 1 out to R = 14; at m = 0 it exceeds 1 at every R, as does cE. The Richardson-extrapolated column overshoots by 2–3% at R = 6, 8, and that overshoot does not collapse between m = 0.05 and m = 0.1 at fixed μR, so it is a lattice artefact of the smallest admissible δ = 2, not evidence against monotonicity. The curves below are measurements on a finite chain, not a theorem.

Where this comes from: claim page · numerics/lattice/petz_thermal_rg.py (output section 2)

Three ways to leave the fixed point Three sketches: with a gap, the conditional mutual information and the recovery error fall with the same power of the gap length; at finite temperature the error is shielded exponentially; along the massive flow both c-functions fall to zero. gap: same power I ~ L_G^-2 -log F ~ L_G^-2 temperature: shielding e^(-2 pi L_B/beta) mass: c falls to 0 c_M c_E
What each deformation does. With JavaScript enabled the three panels below carry the measured points, and the closed forms are evaluated as you move the sliders.

Open a gap

Geometry A(L) | G(LG) | B(L) | C(L), the gap traced out. The line is the closed form evaluated here; the dots are the lattice.

fourteen measured gaps, 0 to 96 sites
L = 6, 8 or 12: the ratio is L-independent

Heat the state

The curve is the KMS closed form at the thermal cross ratio, evaluated here; the dots are the lattice at the same inverse temperature; the dashed lines are the predicted shielding rates 2π and π, anchored at the last measured point, not fitted.

four measured temperatures
the closed form is evaluated at this point

Turn on a mass

Both c-functions against the RG variable μR = mR/2. Everything in this panel is measured.

seven measured masses, 0 to 1.6
R = 6 to 40, in steps of 2

The two-branch law on the lattice

The apparent exponent of −log F(λ) ~ ηp, fitted over η ≤ 0.1, against the values Vardhan, Wei and Zou fitted. A single-branch law is off by factors of 22 to 5000; the two-branch law holds with 1 − 0.15/L, constant in η.

two-branch law on the lattice · numerics/lattice/petz_lattice.py

Claims this module carries

claimstatus
the gap law and linearity off criticalitynumerical
thermal states and exponential shieldingnumerical
the Markov c-function along the massive flownumerical
the two-branch law on the latticenumerical
the adjacent-interval CMI the gap law transportsproved

Four of these five are measurements with the geometry and the chain length they were measured at, not theorems: the closed forms this page evaluates live are exact, the lattice values are not extrapolated to the continuum.