Physics Magnon Gap3 From Jcost

The magnon gap module

An antiferromagnet is a material whose neighboring magnetic moments align in opposite directions. A magnon is a quantized spin wave, a collective excitation of those moments. In a simple one-dimensional model, the energy gap for creating a magnon at the zone boundary is Δ = 2JS√2, where J is the exchange coupling and S is the spin magnitude.

The module Magnon_Gap3_FromJCost does not prove that formula. Its docstring is explicit: the Lean code proves three general facts about a cost function, and nothing specific to magnons. The intended physics appears only in a research note, not in any theorem.

What the code does prove, for any positive real numbers m and e, is that the cost function J(m/e) vanishes when m equals e, and is never negative. It also proves that the golden ratio φ minus 3/2 is positive. These are the three facts bundled into a certificate structure. A certificate is a formal package of these proofs, and the module shows one exists.

The gap between the general facts and the magnon physics is the definition of m and e. In the module, cost is defined as J(m/e) with no reference to exchange coupling, spin, or wave vector. To turn this into a theorem about magnons, one would need to define m and e in terms of the antiferromagnet's own parameters. That definition is absent.

The research note records the intended direction: a gap of the form J(φ) · J_exchange · S · φ^(1/2), which evaluates to about 0.118 · J · 1.27. This is structural, meaning it is a proposed form, not a derived result. The module itself establishes only the three general properties of the cost function, which hold for any positive inputs.

What the module shows, in plain language, is that a certain cost function behaves sensibly: it is zero when its two inputs agree, and it never returns a negative cost. The magnon gap formula remains a research note, awaiting a definition that connects the cost function to the physics.

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

What this page does not claim

This module derives the magnon gap formula Δ = 2JS√2. The research note's structural form is a proved result. The cost function's general properties apply specifically to antiferromagnets.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Physics/Magnon_Gap3_FromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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Derived articles

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