Model I: the coset sigma model

Model I is a field theory on a two-dimensional base, one space and one time direction, whose two fields are geometric structures on the phase plane of a particle moving in one dimension: a metric H, and a real structure K, a linear map of the plane that leaves one line in place and reverses another. These two lines, K’s eigenlines, are the candidate “position” and “momentum” axes. The model’s equations are unchanged by every area-preserving linear map of the phase plane, Born reciprocity ρ among them, but its lowest-energy states are not: ρ carries each vacuum to a different one, and the top-left entry of K’s matrix, K₁₁, which serves as the order parameter, changes sign. The polarization, the angle that sets which line of the phase plane counts as position, can turn at no cost in energy, and in two dimensions such an angle cannot settle on one value by itself (the Mermin–Wagner–Coleman theorem). An added potential, unchanged by ρ and bounded so that it cannot fall without limit, leaves the angle two possible values, which ρ exchanges. With it, model I is the model with which the program meets acceptance criterion 1 in the referees’ wording; its lattice simulation, a computer calculation on a grid of points, holds only for the simplified version with H held fixed.

Stands: coset and hyperbolic sigma models (literature), the action and its symmetries, the vacua, order parameter and Goldstone spectrum, the bounded potential V_b, the frozen-H reduction, the ordered ℤ₂ phase in the Monte Carlo · Died: the naive ρ-preserving potential · Open: a Monte Carlo with H dynamical [O], whether ρ’s SO(2) breaks in the coupled model; folded into a page, where V_b comes from [O] · Superseded: Ising-class breaking in model I, the first version’s Monte Carlo control values, the first version’s unrestricted Mermin–Wagner statement

Map#

Coset and hyperbolic sigma models is the established physics the model is built from: fields valued in a curved target, coset targets and their Goldstone directions, and why an indefinite target metric means a ghost. One group, Sp(2, ℝ), gives the metric a target with a positive metric, the hyperbolic plane, and the real structure one with an indefinite metric, two-dimensional de Sitter space, which would carry a ghost. The page also states the theorem by which a compact continuous symmetry cannot break spontaneously in two dimensions, and the hyperbolic models that evade it. It comes first because every later page uses it.

Model I: action and symmetries sets up the phase plane (ω, ρ, the real structure K, the neutral metric η = ωK and the metric H), then gives the Euclidean Lagrangian term by term: a hyperbolic kinetic term for H, a kinetic term for K contracted with H so that it carries no ghost, and a potential that vanishes when the two fields are compatible. The action is invariant under Sp(2, ℝ), hence under ρ, and under a separate flip K → −K. That is criterion S1, machine-checked.

Model I vacua, the order parameter and the Goldstone spectrum finds the lowest-energy states (criteria S2 and S3): the compatible pairs, the upper half-plane times a circle of real structures K(φ). ρ moves every vacuum, only ±1 fix one, and O = K₁₁ changes sign. Three of four directions are flat, one of them the compact polarization angle, which parametrizes Λ(1). The page closes with the twisted element ρ∘(K → −K), the reason a potential is needed, and the link to τ’s circle, which is consistency, not a test.

Obituary: the naive ρ-preserving potential records the first potential added to make the polarization definite, −μ[Tr(K K_ref)]². It was tested only with H frozen; round 4 showed it falling to −8.9 × 10⁶ μ along a boost family once H moves, with a saddle where the vacuum was meant to be. It sits before its repair, so the repair reads as the answer to a measured failure.

The bounded potential V_b is that repair: the naive form divided by 2 Tr KᵀK, bounded between −μ and 0 by Cauchy–Schwarz and stable with all fields dynamical. One flat direction remains that no symmetry protects, and the sources disagree about it; version 2 governs. Where the potential comes from is open.

The frozen-H reduction states the scope of the lattice result: with H held at 1, model I becomes a two-dimensional spin model of the polarization angle. The page explains why the scope had to be stated, how spontaneous the breaking in the reduction is (the potential fixes a pair of axes, the vacuum chooses between them), why “Ising class” became “ordered ℤ₂ phase”, and where the simulation with H dynamical stands.

Monte Carlo: an ordered ℤ₂ phase is the numerical evidence for criterion S4 in the reduction: the lattice model and the Metropolis method, the three runs with error bars from independent seeds, hot-start trapping, the referee’s independent simulation, and the first version’s control values that round 4 replaced.

Open: a Monte Carlo with H dynamical is the first of the toy work’s open ends [O] (#11 §14 item 11): the ordered phase has been simulated only in the reduction, and whether it survives when H fluctuates as well has not been computed. The page sets out what the frozen-H simulation shows, why round 4 required the result to be stated for the reduction, and the classical check that exists instead, with all four field directions free: Hessian eigenvalues 0, 0.764, 2, 5.236, the zero belonging to a flat line that no symmetry protects. It then records how the full model differs from the reduction and what would close the item; no source says how the simulation would be set up.

Open: does ρ’s SO(2) break in the coupled model? closes the subsection with the question the reduction leaves: whether the full model, whose metric sector is a hyperbolic sigma model, orders ρ’s compact SO(2) without a potential.

Only the first page is literature [T]. The rest are the program’s claims, from papers/reciprocity-toy-models.md §2 and §4 and the round-4 review, checked in verify/toy_coset.py and verify/toy_s4.py (the toy-model scripts).

Boundaries#