cft-cmi

Markov c-function along the massive flow

Status: numerical — measured, with an error bar and a convergence order; not a theorem.

Area: Lattice, thermal, massive and gapped geometries (O7, O9, O10) · Card type: result · Knowledge-base id: rg-c-function

Statement

For the massive (staggered-mass) Dirac chain, c_M(R) := 3R^2 I_{R-2}(2)/4 and c_E(R) := 3R (S(R+2) - S(R-2))/4 satisfy c_M >= c_E at every computed point, and along the flow (mu R from 0 to infinity, mu = m/2) both fall monotonically from c = 1 towards 0 (dc_E/dR < 0 and dc_M/dR < 0 at every point with m > 0; numerics/lattice/petz_thermal_rg.out l. 393-549). The ratio c_M/c_E behaves like m R only in a scaling window (correlation length xi = 2/m >> 1 site and mu R >> 1; the free-field prediction from S(R) - S(infinity) ~ A e^{-2 mu R}); on the lattice at fixed m the ratio (c_M/c_E)/(mR) does not tend to 1: for m = 0.4 it is 1.196, 1.077, 1.020, 1.003, 0.992, 0.972 at R = 6, 10, 16, 20, 24, 40 and still falling, and at R = 40 it is 0.825 for m = 0.8 and 0.563 for m = 1.6 (plateaus f(0.8) ~ 0.83, f(1.6) ~ 0.56), while for small m (0.05, 0.1, 0.2) it is still above 1 and decreasing at R = 40 (1.36, 1.15, 1.05). This is consistent with c_M/c_E ~ m R in the continuum scaling limit (m -> 0 at fixed mu R >> 1), which the data do not reach; rigor/thermal_rg_gap_summary.md:73 quotes only R <= 16 at m = 0.4.

How to verify

numerics/lattice/petz_thermal_rg.out, RG section (l. 393-549; columns c_E, c_M, c_M/c_E, (c_M/c_E)/(mR), dc_E, dc_M per mass); petz_thermal_rg.py l. 429-470 (cfun_scan); rigor/thermal_rg_gap_summary.md l. 60-80.

Evidence

Related claims

None recorded.

Review

Review record: passed 2026-09-23 by REF-DEP-CFT-B4.

The referee's report and the card's event log stay in the private knowledge base; only the verdict, the date and the referee are published here.

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