Model I: the coset sigma model on The Broken-Born program

Coset and hyperbolic sigma models

Mon, 01 Jan 0001 00:00:00 +0000

A sigma model is a field theory whose field takes values, at each point of spacetime, in a curved space called the target, such as a circle, a sphere or a hyperbolic plane; the action charges a cost whenever neighbouring points take distant values in the target. In a coset sigma model the target is a group divided by a subgroup, so the group acts on the target as a symmetry and every constant field is a lowest-energy state. Two results from this literature shape model I, the program’s coset model of phase-space structures. The target’s metric must be positive, or the kinetic term carries a ghost, a mode of negative energy. And in two dimensions a compact continuous symmetry cannot break spontaneously, while the non-compact symmetry of a hyperbolic target can.

Model I: action and symmetries

Mon, 01 Jan 0001 00:00:00 +0000

Model I’s action is the integral, over a flat two-dimensional base, of a Lagrangian with three terms: two kinetic terms, which cost action when the fields vary from point to point, and a potential that vanishes exactly when the two fields are compatible. The fields are geometric structures on the phase plane of one particle moving in one dimension: a metric H, parametrized by a point of the upper half-plane, and a real structure K, whose two eigenlines are the candidate position and momentum axes. The action is unchanged by every symplectic (area-preserving) linear map of the phase plane, which includes Born reciprocity ρ, and by the separate flip K → −K. That invariance is criterion S1, checked symbolically, and it supplies the “genuine Noether symmetry” that acceptance criterion 1 asks for.

Model I vacua, the order parameter and the Goldstone spectrum

Mon, 01 Jan 0001 00:00:00 +0000

The vacua of model I are its lowest-energy configurations: constant fields H and K that are compatible, so that the potential vanishes. They form a three-dimensional family, the upper half-plane of metrics times a circle of real structures, and Born reciprocity ρ carries each vacuum to a different one, so no vacuum is symmetric (criterion S2). The entry K₁₁ of the real structure takes opposite signs in a vacuum and in its image, which makes it an order parameter (criterion S3). Three of the four directions in which the fields can vary are flat. One of them is the polarization angle: moving along it turns the line of the phase plane that counts as position, at no cost in energy. Checkpoint #11 identifies that circle with the one on which its dark-energy field τ lives, as a matter of consistency, not as a test.

Obituary: the naive ρ-preserving potential

Mon, 01 Jan 0001 00:00:00 +0000

The naive ρ-preserving potential was the term −μ[Tr(K K_ref)]², added to model I so that its circle of vacua would reduce to two and the breaking of Born reciprocity could survive in two dimensions. It is symmetric under ρ and gives the two intended vacua in the frozen-H reduction, where the metric H is held at 1 and K stays on its circle of angles. Once H is allowed to vary it is unbounded below: along a family of boosted configurations the compatibility term stays at zero while the potential falls without limit, and the intended vacuum is a saddle. The worklog’s first pass of criterion S4 and the first version of the toy-model note rested on it. The round-4 referee found the defect, the adjudication upheld it, and the potential survives only as the counterexample that the bounded potential V_b answers.

The bounded potential V_b

Mon, 01 Jan 0001 00:00:00 +0000

V_b is the repaired ρ-preserving potential of model I: the naive potential divided by a normalizing factor, 2 Tr KᵀK, which keeps it between −μ and 0 for every real structure K. Like the naive form it is unchanged by Born reciprocity ρ and by the flip K → −K, and it reduces the circle of real structures to the pair ±K_ref. Unlike the naive form, it leaves the vacuum a stable minimum when the metric H is free to vary: the boosted configurations along which the naive form fell without limit now cost energy. The round-4 referee proposed it, version 2 of the toy-model note adopted it, and checkpoint #11 records acceptance criterion 1 as met with it, in the referees’ wording. One flat direction remains that no symmetry protects, and where such a potential comes from is not derived.

The frozen-H reduction

Mon, 01 Jan 0001 00:00:00 +0000

The frozen-H reduction is model I with its metric field held fixed at H = 1, so that the only remaining field is the polarization angle φ of the real structure K(φ). In that form the model is a two-dimensional spin model of one angle, with a stiffness term and a ρ-preserving potential, and it can be simulated on a lattice. The ordered ℤ₂ phase the simulation finds is a result about this reduction; with H dynamical, ordering is not established. Round 4 required the text to say so, after showing that the first potential tried was stable only in this reduction. It also replaced the claim of “Ising-class” breaking with “an ordered ℤ₂ phase”, because data at one temperature establish order but not a universality class.

Monte Carlo: an ordered ℤ₂ phase

Mon, 01 Jan 0001 00:00:00 +0000

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.

Open: a Monte Carlo with H dynamical

Mon, 01 Jan 0001 00:00:00 +0000

A Monte Carlo with H dynamical is a lattice simulation of model I in which the phase-space metric H varies from point to point and fluctuates together with the real structure K, instead of being held at the constant value H = 1. The program’s simulation is of the simpler model with H held fixed, the frozen-H reduction, which keeps only the polarization angle, the angle that sets which line of the phase plane counts as position. With the bounded potential V_b added, that angle stays near one of the two values that Born reciprocity ρ exchanges: an ordered ℤ₂ phase. Whether the order survives when H fluctuates as well has not been computed. Round 4 required the lattice result to be stated for the reduction only, after finding that the first potential tried, stable in the reduction, fell without limit once H was allowed to move. For the full model there is a classical check instead: the vacuum is a minimum of the potential with all four field directions free, though along one of them the potential is flat and no symmetry keeps it so. Checkpoint #11 lists the simulation as open, and no source says how it would be set up.

Open: does ρ's SO(2) break in the coupled model?

Mon, 01 Jan 0001 00:00:00 +0000

The open question is whether model I, with its metric field H free to vary and no added potential, spontaneously breaks the rotation group SO(2) of the phase plane, the circle of symmetries that contains Born reciprocity ρ. Breaking it would mean that the vacuum settles on one polarization angle despite quantum fluctuations. In two dimensions a compact continuous symmetry cannot break on its own (the Mermin–Wagner–Coleman theorem), so the polarization angle by itself cannot order. But model I’s metric sector is a hyperbolic sigma model, a known case in which a non-compact symmetry does break in two dimensions, and the two sectors are coupled. Round 4 required the toy-model note to restrict the theorem to the compact SO(2) and to state the coupled case as open; the note lists it as not shown.