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

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.

Status: Open — untagged — raised by the round-4 referee (conceptual point C2), upheld as a literature point in the adjudication, and listed under “Not shown” in version 2, §6. Registered at open problems.

The question#

Version 2 of the toy-model note: “Whether ρ’s SO(2) stays unbroken in the full coupled model is open.” (§4). Its list of what is “Not shown” includes “whether ρ’s SO(2) breaks in the coupled model without a potential” (§6).

ρ’s SO(2) is the group of rotations of the (x, p) plane. Among the symplectic maps they are the ones that leave the metric H = 1 unchanged, the stabilizer of the point Z = i of the upper half-plane [T] (the Siegel domain). Acting on the real structure, they turn K around its circle of vacua K(φ), and ρ is the quarter-turn among them. The worklog calls this circle group “the U(1) stabilizer of H that rotates K” (‘Chunk 2 results’); SO(2) and U(1) are the same group [T]. If SO(2) broke spontaneously, the vacuum would hold one definite angle φ, one definite polarization. “Without a potential” means with model I’s own three terms only (Model I: action and symmetries).

What is known#

The compact angle by itself cannot order#

“In 1+1 the compact angle φ cannot order: Mermin–Wagner [13, 14] applies to the compact SO(2) containing ρ.” (§4). In two dimensions the long-wavelength fluctuations of a compact angle grow without limit and wash out any preferred value [T]. The lattice simulation of the angle with H frozen shows this: without the potential the order decays with lattice size, “⟨|M|⟩ = |⟨e^{iφ}⟩| = 0.8369 ± 0.0007, 0.8025 ± 0.0039, 0.7736 ± 0.0038” at L = 16, 32, 64, read as “No true order” (§4; verify/toy_s4.py, see the toy-model scripts; Monte Carlo: an ordered ℤ₂ phase).

The metric sector can break#

“The non-compact half-plane sector is a hyperbolic model, where symmetry breaking in 2D does occur [15].” (§4). The round-4 referee’s point C2: “Model I’s group Sp(2,R) is non-amenable and its H-sector is the SO(1,2)/SO(2) hyperbolic sigma model”, for which Duncan, Niedermaier and Seiler report spontaneous breaking in two dimensions, “For D=2 this shows that the systems evade the Mermin-Wagner theorem” (review/round4/referee/audit-notes.md, Step 3). The literature is explained at coset and hyperbolic sigma models.

The two sectors are coupled#

In model I the two sectors do not decouple. K’s kinetic term is contracted with H, and the compatibility potential ties K to H (the K kinetic term). The theorem speaks about a compact symmetry, and the hyperbolic result about a non-compact one; the adjudication accordingly restricts the theorem and leaves the coupled case open: “The Mermin–Wagner statement must be restricted to the compact SO(2) containing ρ. Whether that SO(2) is broken in the coupled model is open.” (review/round4-adjudication.md, ‘Rulings’, “Mermin–Wagner vs [11]”; [11] is the first version’s number for reference [15]).

How the question arose#

The worklog anticipated the issue when it fixed criterion S4: “A continuous compact one cannot (Coleman–Mermin–Wagner), though there is a claimed exception for non-amenable groups, to be checked.” (‘Success criteria’). Its forecast: “S4 will pass only for a discrete reduction (the symmetry group cut down to the group generated by ρ) or via the non-amenability exception. The continuous U(1) of phase-space rotations that contains ρ would be restored in 1+1.” (‘Predictions’).

After the first construction it wrote: “S4 is the open question. The breaking is continuous, so in 1+1 the compact circle mode should restore ⟨K⟩ (Coleman–Mermin–Wagner) unless the non-amenable directions change that. Waiting on the S0 literature check.” (‘Chunk 1 results’). The prior-art survey retrieved the hyperbolic-model papers [S], and the worklog listed two ways forward: a discrete reduction by a potential, “Or the non-amenability route (Duncan–Niedermaier–Seiler), whose applicability to this target is unchecked.” It took the first (Obituary: the naive ρ-preserving potential). The assessment records the second as “Non-amenability not needed, not tested” (‘Assessment against the predictions’).

Round 4 returned to it. The referee’s point C2 continues: “The statement must be restricted to the compact SO(2) containing rho, with an argument.” (Step 3). The adjudication upheld the point from the literature, and revision 1 required version 2 to “Restrict Mermin–Wagner to the compact SO(2)” and to cite the hyperbolic-model papers “for the hyperbolic H-sector, noting the open question”. Version 2, applied on 2026-10-20, does both (§4, §6).

Superseded: the first version’s unrestricted theorem#

The first version of the note (commit d9a06ac) stated the theorem for continuous symmetries in general, with non-amenable groups as an exception, and then applied it to the whole of model I’s continuous breaking without that exception. Its §1: “Continuous symmetries cannot break spontaneously in two dimensions [9, 10]. Non-amenable groups are an exception [11].” Its §4: “In the exactly symmetric model the continuous breaking is forbidden [9, 10].” In that numbering [9, 10] are Mermin–Wagner and Coleman, and [11] is the hyperbolic-model pair, now [13, 14] and [15].

The referee found the two sections at odds: §4’s statement “ignores [11] for model I’s own (non-amenable) Sp(2,R)” (Step 1). Version 2 replaces it with the restricted statement quoted above, and the change is listed among the superseded statements.

What the note does instead#

Version 2 obtains a definite polarization without answering the question, in two steps. It adds a potential: “To get a definite polarization we add a potential that is invariant under ρ and under K → −K, and reduces the U(1) to a discrete group.” (§4; the bounded potential V_b). And it simulates the angle with H frozen, where the reduced symmetry orders: “an ordered ℤ₂ phase in the frozen-H reduction in 1+1” (§6; the frozen-H reduction). The program’s one ordered phase is therefore obtained with an explicit potential in a reduced model. The coupled model without a potential has been neither simulated nor analysed.

The worklog’s third unpredicted finding was stated for 1+1: “In 1+1, ‘only ω rigid’ is not enough for a definite polarization. A ρ-preserving potential on the polarization circle is required.” (‘Assessment against the predictions’). Version 2 keeps the coupled case apart, listing “whether ρ’s SO(2) breaks in the coupled model without a potential” as not shown.

What the sources leave unsaid#

The sources give no method for settling the question and no expectation for its answer. The note lists it beside “a Monte Carlo of model I with H dynamical” (§6), which #11 §14 item 11 also keeps open [O] (Open: a Monte Carlo with H dynamical); the note lists the two as separate items. The worklog’s separate remark that the continuous breaking is allowed in 3+1 dimensions, where the theorem does not apply, is an uncomputed remark recorded at the polarization as a Goldstone angle; the note covers only 1+1 with d = 1.

Sources#

  • papers/reciprocity-toy-models.md §1, §4, §6 — the theorem and its exception, the restricted statement, the open question and the “Not shown” list
  • papers/reciprocity-toy-models.md v1 (commit d9a06ac) §1, §4 — the unrestricted statement, now superseded
  • review/round4-adjudication.md ‘Rulings’ (“Mermin–Wagner vs [11]”), ‘Revisions’ 1 — upheld (literature); the required restriction
  • review/round4/referee/audit-notes.md Step 1, Step 3 — the hyperbolic-model citation and its non-application in v1; conceptual point C2
  • notes/toy-lagrangian.md ‘Success criteria’, ‘Predictions’, ‘Chunk 1 results’, ‘Chunk 2 results’, ‘Chunk 3 results’, ‘Assessment against the predictions’ — criterion S4, the forecast, the framing as “the open question”, the U(1) stabilizer, and the non-amenability verdict
  • papers/broken-born-checkpoint-v2.md §14 item 11 — the Monte Carlo with H dynamical kept open [O]
  • verify/toy_s4.py — the run without the potential