Monte Carlo: an ordered ℤ₂ phase

The Monte Carlo is a lattice simulation of the frozen-H reduction of model I: a square grid of angles φ, one per site, with an energy that is lowest when neighbouring angles agree and, when the ρ-preserving potential is switched on, lowest at the two angles 0 and π. Random local changes are accepted or rejected by the Metropolis rule at a fixed temperature, and the lattice average m of cos φ, which changes sign under ρ, shows whether one of the two vacua has been singled out. With the potential the mean of |m| stays at 0.954 on every lattice size, an ordered ℤ₂ phase. Without it the order decays slowly with size, as a continuous symmetry in two dimensions requires, and at high temperature it falls as the inverse of the lattice size. This is criterion S4 for the reduction, with error bars from four independent seeds, adopted after round 4 found the first control numbers under-sampled.

Status: Alive, for the frozen-H reduction only — [V] (the worklog records “S4 passes only with the τ-type potential [V]”, the naive form, which at H = 1 gives the same lattice energy; verify/toy_s4.py, gated; the toy-model scripts) — the ordered-phase values and the potential-free decay agree with the round-4 referee “to about 1%”; the control values were corrected in version 2 (review round 4).

The lattice model#

The simulated energy is model I’s polarization angle at H = 1 on a square lattice (toy-model note §4):

E = −J Σ⟨ij⟩ cos(φ_i − φ_j) − h Σ_i cos 2φ_i

The first sum runs over neighbouring sites and is the lattice form of the angle’s stiffness; the second is the bounded potential V_b at H = 1, “up to a constant and normalization” (§4). The lattice has L × L sites with periodic boundaries: the neighbour shifts in verify/toy_s4.py wrap around the edges. The runs use J = 1, h = 1 with the potential and h = 0 without it, temperature T = 0.6, a control at T = 3.0, and L = 16, 32 and 64. The lattice coupling J is unrelated to the J of #11’s postulate P2′.

Born reciprocity acts on the lattice as φ → φ + π at every site. The order parameter is “m = ⟨cos φ⟩” (§4), the lattice average of cos φ in one configuration. Since cos φ is the entry K₁₁ of K(φ), m is the lattice average of model I’s order parameter O = K₁₁, and it changes sign under ρ (model I vacua). The script records |m| and averages it over the run, ⟨|m|⟩, because in a finite lattice the sign of m can flip over a long run, which would average m itself to zero [T]. Without the potential the symmetry is the whole circle of shifts, and the script uses “the U(1) magnetization |mean exp(i phi)|” (verify/toy_s4.py), written ⟨|M|⟩ = |⟨e^{iφ}⟩| in the note.

The algorithm#

The Metropolis algorithm samples configurations with the Boltzmann weight e^{−E/T} [T] (Metropolis et al. 1953†). Each step proposes a random change; a change that lowers the energy is accepted, and one that raises it by ΔE is accepted with probability e^{−ΔE/T}. In verify/toy_s4.py each proposal shifts an angle by an amount drawn uniformly between −1 and 1. “Checkerboard” means the sites are split into two interleaved sublattices like the squares of a chessboard; every neighbour of a site lies on the other sublattice, so a whole sublattice can be updated at once [T]. One sweep updates both.

Version 2: “Checkerboard Metropolis, cold start (φ = 0), L = 16, 32, 64. Means ± standard errors over four independent seeds.” (§4). In the script each run discards 1,500 sweeps for equilibration and then measures over 4,000 (its defaults “therm=1500, sweeps=4000”). Each quoted value is the mean of four runs with different random seeds, and its error bar is the standard error of that mean: the spread of the four results divided by √4. The seeds are fixed, so the whole computation, error bars included, is reproducible: “Fixed seeds, so the run is deterministic.” (docstring; the Monte Carlo error-bar rule).

What the diagnostics show [T]#

• Size dependence of ⟨|m|⟩. In an ordered phase ⟨|m|⟩ approaches a nonzero constant as L grows. In a disordered phase m averages L² nearly independent angles, so its typical size falls as 1/√(L²) = 1/L.
• The Binder cumulant. The script computes U = 1 − ⟨m⁴⟩/(3⟨m²⟩²). If m takes only two opposite values ±m₀, then ⟨m⁴⟩ = ⟨m²⟩² and U = 2/3. If m is spread as a Gaussian about zero, as in a disordered phase, then ⟨m⁴⟩ = 3⟨m²⟩² and U = 0. The script’s check reads a value “near 2/3” as “an ordered Z2 phase”.
• Spin waves without the potential. In two dimensions the XY model at low temperature has no true long-range order, but its order decays only as a power of distance. In the Gaussian, or spin-wave, approximation the magnetization of an L × L lattice falls as L^{−T/4πJ}. Between L = 16 and L = 64 that predicts the ratio (64/16)^{−T/4πJ}, which the note evaluates as 0.936 at T = 0.6 and J = 1. The script computes the same estimate from η = T/2π at J = 1, the power with which correlations decay with distance.

Results#

Version 2’s table (§4), reproduced by the gate:

With the potential: an ordered ℤ₂ phase#

The gate prints ⟨|m|⟩ = 0.9543 ± 0.0001, 0.9542 ± 0.0001 and 0.9542 ± 0.0000, with Binder cumulants 0.6666, 0.6667 and 0.6667, at L = 16, 32 and 64 (verify/toy_s4.py). The note’s “± 0.0001 at every size” is an upper bound at L = 64, where the printed error rounds to 0.0000 (flag 13 in review/wiki-build-flags.md); the gate accepts any standard error below 0.0002 for this row. The script passes four checks on this row: “ordered: <|m|> > 0.8 at every size”; “size-independent within errors (|m64 - m16| < 3 sigma + 0.002)”, with the difference printed as −0.0001 and sigma as 0.0001; “Binder cumulant near 2/3 (an ordered Z2 phase)”; and the quoted values.

⟨|m|⟩ does not fall as the lattice grows fourfold, and the Binder cumulant sits at its ordered value: the shift by π is spontaneously broken, and the lattice stays near one of the two angles 0 and π instead of visiting both. Version 2 states the limit of the claim: “We call the first row an ordered ℤ₂ phase. We did not measure critical exponents. Symmetry-breaking fields in the planar model are treated generally in [17]; the ordered phase at these parameters is established by the simulation itself.” (§4).

Reference [17] is J. V. José, L. P. Kadanoff, S. Kirkpatrick and D. R. Nelson, Phys. Rev. B 16, 1217 (1977), which version 2 keeps as “Background on symmetry-breaking fields in the planar model; the ordered phase used here is established by our own Monte Carlo, not by appeal to this reference.” The first version had cited it, as [12], for “the two-fold anisotropic planar model, whose ordered phase is Ising-like” (v1 §4). The pre-registration named it the likeliest reference “cited for more than it says” (review/round4-preregistration.md). The referee found that its abstract treats “hexagonal and cubic crystalline fields; no mention of the two-fold/Ising case”, and with the full text paywalled the specific claim could not be checked (review/round4/referee/audit-notes.md, Step 1). The adjudication recorded “[12]’s specific claim is unverifiable”.

The control at T = 3: the sampler leaves the ordered start#

Every run starts from the ordered configuration φ = 0. At T = 3.0 the values fall with size: the gate passes “disordered: <|m|> scales as 1/L (m16/m64 within [3.3, 4.8], expected 4)” with the ratio 4.209. A disordered phase gives 1/L, so the sampler does leave its ordered start when the temperature puts the system in a disordered state. The worklog: “The ordered start is a choice; the high-T control shows the sampler does leave it when the phase is disordered.” (‘Chunk 3 results’); the script’s comment on its start reads “ordered start; equilibration tested by the high-T control”.

Without the potential: no true order#

With h = 0 the magnetization falls with size, slowly: the gate passes “no true order: the U(1) magnetization falls with size by more than 3 sigma”, printing the ratio 0.9243 against the spin-wave estimate 0.9360. This is the lattice form of the Mermin–Wagner–Coleman theorem: a continuous compact symmetry in two dimensions does not order.

The referee checked the spin-wave formula on the lattice: “lattice Gaussian sum gives M(64)/M(16) = 0.9359 vs closed form (64/16)^(-T/4piJ) = 0.9360” (Step 2(c); review/round4/referee/out_sec4_extra.txt). The data fall below the estimate, which the referee read “as expected from stiffness renormalisation (referee fitted exponent -0.0628 vs bare -0.0477)”. At nonzero temperature, fluctuations beyond the Gaussian approximation soften the effective stiffness, so the order decays faster than the bare spin-wave exponent says [T].

Start conditions: hot starts trap#

A hot start begins from random angles. The referee ran both starts. At L = 16 and 32 they agree: 0.9541 from both at L = 16, and 0.9542 (hot) against 0.9543 (cold) at L = 32, with Binder cumulants 0.6666 and 0.6667 (review/round4/referee/out_xy_mc.txt). At L = 64 the hot start “did not equilibrate in 6000 sweeps: <|m|> = 0.0593 (U=0.536)” (Step 2(c)). Diagnostic runs at L = 64 (review/round4/referee/out_xy_trap.txt):

The cold start ends at the lowest energy. The referee’s reading: “Metastable domain stripes with higher energy; equilibrium is ordered.” A random start in an ordered phase breaks up into domains of both signs, with angles near 0 in some and near π in others, separated by walls that cost energy [T]. The walls move slowly, and on a periodic lattice a wall that wraps all the way around the system can persist for a long time [T].

The referee also cautioned: “Manuscript does not state its start condition; results from a cold start alone would be partly by construction.” The ruling on start conditions, “Hot starts at L = 64 trap in metastable domains; start conditions not stated”, was accepted: “Cold starts are standard for ordered-phase measurements, but they must be stated.” (review/round4-adjudication.md, ‘Rulings’, the row “MC start”, MC standing for Monte Carlo). Version 2 states them: “Hot starts at L = 64 coarsen slowly through metastable domains, so ordered-phase values are reported from cold starts.” (§4); the script’s docstring: “All runs use a cold start (phi = 0).”

The referee’s independent simulation#

The round-4 referee wrote its own checkerboard Metropolis, with its own seeds, longer runs and error bars from binning a single run (review/round4/referee/ref_xy_mc.py), and compared it with the first version of the note (Step 2(c)):

The adjudication: “The ordered-phase values (0.954) and the h = 0 decay agree with the referee to about 1%.” (‘Rulings’, “MC control”). For h = 0 the referee noted: “Values agree at L=16; L=32/64 differ by ~0.006-0.009 (3-4 binned sigma, binned errors likely underestimated for Metropolis autocorrelation at L=64).” Successive Metropolis sweeps are correlated, so error bars computed from blocks of one run come out too small when the blocks are short compared with the correlation time [T]; independent runs avoid that. The values this wiki cites for h = 0 are version 2’s four-seed ones in the table above.

Superseded: the first version’s control values#

The first version (v1 §4, commit d9a06ac) and the worklog quoted the high-temperature control as “0.078 → 0.033 → 0.014”, from fixed seeds and without error bars, and the run without the potential as “0.838 → 0.798 → 0.776, ratio 0.925” (‘Chunk 3 results’).

The referee confirmed the qualitative reading (disordered), but not the numbers: “the quoted sequence scales as L^-1.239 vs the 1/L expected and found (referee exponent -1.0069); quoted L=16 value is 1.238x the referee’s (0.078 vs 0.0630+-0.0004). Minor: likely under-sampled” (Step 2(c)). The adjudicator reran the control with the script’s own run function (review/round4/adjudicator/ctrl.py). At L = 16 and T = 3, seed 116 gave 0.0777, seeds 1, 2 and 3 gave 0.0687, 0.0639 and 0.0626, and a long run of 40,000 sweeps gave 0.0653; long runs at L = 32 and 64 gave 0.0325 and 0.0164 (review/round4/adjudicator/ctrl.out). The ruling, upheld:

ctrl.py: my L = 16 value from seed 116 was a short-run fluctuation. Other seeds give 0.063–0.069, and long runs give 0.0653 / 0.0325 / 0.0164, clean 1/L. Fixed seeds gave reproducibility, not accuracy; the numbers lacked error bars.

The current values are version 2’s four-seed means, 0.0653 ± 0.0043, 0.0338 ± 0.0013 and 0.0155 ± 0.0004, which the gate reproduces. The adjudicator’s long runs differ from them at L = 32 and L = 64 (0.0325 against 0.0338, and 0.0164 against 0.0155); the sources do not reconcile the two sets, and this wiki cites version 2’s (conflicts between sources). The values without the potential changed too: version 2’s 0.8369, 0.8025, 0.7736 and ratio 0.924 replace the first version’s. That first-version ratio appears as 0.925 (v1 and the worklog), 0.926 (the referee’s notes) and 0.9260 (out_sec4_extra.txt); the last two were recomputed from the rounded values.

The pre-registration had predicted “at most one surviving numerical indictment”, likeliest in “a normalization or convention: the spin-wave exponent, the 8π(m+n+1) coefficient, or the Δ sign convention”. The outcome was scored “Hit on count, miss on location”: “One: the MC control values, a statistical problem. Every normalization and sign I listed was confirmed” (‘Pre-registration scored’). Revision 1 asked for “start conditions, equilibration checks, and error bars from independent seeds; rerun the control”, and round-4 lesson 2 became a rule in the root CLAUDE.md: “Monte Carlo numbers need error bars from independent runs. A fixed seed makes a number reproducible, not accurate.” (the Monte Carlo error-bar rule). The replacement is listed among the superseded statements.

Scope#

The simulation is of the reduction only: “H is not simulated” (§4). The worklog’s run used the same lattice energy under the name “τ-type potential”, the naive form. At H = 1 the naive form and V_b differ only by an overall factor, and on the lattice either becomes −h cos 2φ up to a constant, so the lattice results stand for both; the naive form fails only once H is dynamical. A Monte Carlo of model I with H dynamical is open (Open: a Monte Carlo with H dynamical), and so is whether the coupled model orders ρ’s SO(2) without a potential (Open: does ρ’s SO(2) break in the coupled model?). The result counts toward acceptance criterion 1 as “an ordered ℤ₂ phase in 1+1 in the frozen-H reduction (Monte Carlo)” (#11 §14 item 11).

Sources#

• papers/reciprocity-toy-models.md §4, ref [17] — the method, the three runs and their readings, the start conditions, quoted above
• papers/reciprocity-toy-models.md v1 (commit d9a06ac) §4 — the first-version values and the Ising-like citation
• verify/toy_s4.py — the gated simulation, its parameters and checks, which reproduce every quoted value
• review/round4-adjudication.md ‘Rulings’ (MC control, MC start), ‘Pre-registration scored’, ‘Revisions’ 1, ‘Lessons’ — the rulings on the control values and on the start conditions; the scoring; the error-bar lesson
• review/round4/adjudicator/ctrl.py, review/round4/adjudicator/ctrl.out — the seed-116 value and the long runs
• review/round4/referee/audit-notes.md Step 1, Step 2(c) — the citation check of [17]; the referee’s independent simulation, the hot-start trapping and the spin-wave check
• review/round4/referee/out_xy_mc.txt, review/round4/referee/out_xy_trap.txt, review/round4/referee/out_sec4_extra.txt, review/round4/referee/ref_xy_mc.py — the referee’s runs, trap diagnostics, spin-wave sums and code
• review/round4-preregistration.md ‘Predictions’ — the predicted numerical indictment and the citation flag
• notes/toy-lagrangian.md ‘Chunk 3 results’ — the first-version values, now superseded, and the role of the control
• CLAUDE.md ‘Method’ — the error-bar rule