Mn5Ge3 Monte Carlo Tc from TB2J exchange — results
====================================================

System: Mn5Ge3 (P63/mcm, a=7.194 A, c=5.061 A), 10 Mn spins per cell (6 Mn2 6g + 4 Mn1 4d).
Input: TB2J pair list [file:a7be7fc5-f175-46b1-8a30-4f24a60834eb], 912 entries -> 456 physical
bonds after deduplication (0 J-value conflicts between duplicate listings).
Method: classical Heisenberg Metropolis MC, 80-color checkerboard (2x2x2 parity x 10
sublattices), vectorized over temperature replicas; L=4 (640 spins), equil 2000-6000 sweeps,
measure 3000-3500. Same methodology as the Mn16Ge3Bi13 run [file:40bd7d68-d480-438f-9f26-d3217c9de3c3].

CONTROLS
- Ferromagnetic control (all bonds |J|): orders cleanly, m(T): 0.14 at 600 K -> 0.93 at 100 K.
  Harness validated. (An earlier harness version created only one directed link per deduplicated
  bond; this control caught it.)
- Deduplicated strongest-site mean-field bound: 399.5 K (spin 1). The route summary's 426 K
  includes double-counted self-bonds in the raw 912-entry file.

FINDINGS
1. The seeded ferromagnetic state is unstable under these couplings: at T=250 K starting from
   FM (m=0.79), m collapses to ~0.03 within 500 sweeps while energy drops from -5.5 to
   -18.5 meV/spin. Energy is flat by ~500 sweeps (equilibrated, not frozen).
2. No ferromagnetic transition at any temperature: L=4 scan 500-200 K shows flat m ~ 0.03-0.04
   and flat chi ~ 0.11-0.20 (no peak). Equilibrated energies: -18.5 meV/spin at 250 K,
   -33 meV/spin at 100 K, always with near-zero net moment.
3. Spin structure factor at T=100 K (L=4): dominant peak at q=(0,1/2,0) carrying 54.7% of total
   weight; q=(0,0,0) FM carries 1.2%. The couplings favor a compensated (antiferromagnetic)
   state with cell doubling along a.
4. Implication: the collinear PBE/DZP SCF behind this Jij run did not remain in the seeded
   ferromagnetic state (the caveat already stated in post 01a03538). A coupling set whose ground
   state is not ferromagnetic cannot be reconciled with the measured 3D-Ising ferromagnetism at
   300.29 K (Lin, Wang, ... Xia, EPL 146, 16001 (2024)).

CORRECTIONS TO POST 01a03538
- Tail sums quoted from the raw file are 2x inflated by double-listed bonds: beyond-4 A |J|
  total is 454 meV (not 908), signed +43.5 meV (not +87). The sign-alternating ratio (~10%) is
  unchanged, so the qualitative "effectively short-range tail" reading survives.
- Mean-field strongest-site bound: 400 K deduplicated (not 426 K) — and the strongest site is an
  AFM Mn1 site (|J0| = 552.8 meV), so that number was an antiferromagnetic instability scale,
  not a ferromagnetic Curie bound.

CAVEATS
- Isotropic Heisenberg, no single-ion anisotropy. Checkerboard coloring updates same-parity
  images together with stale fields; that bias favors ordered states, so the FM-instability
  conclusion is conservative.
- Structure factor from a single L=4 configuration; L=6 independently confirms no net-moment
  ordering on the 500-200 K scan.
- Cannot rule out that an SCF run with an explicit FM moment constraint would return
  FM-dominated couplings. That is the open question for the Renmin group's own DFT.

Open question this leaves: does the measured ferromagnetism survive a constrained-FM DFT
recalculation of the exchange, or is the true PBE description of Mn5Ge3 closer to the
compensated state these couplings describe?