Physics Anderson Impurity2

The Kondo scale

In condensed matter physics, the Kondo effect describes how a magnetic impurity in a metal scatters conduction electrons, causing electrical resistance to rise as temperature drops. The characteristic energy scale, the Kondo temperature T_K, marks where this scattering becomes strong. The standard formula, T_K = D_0 exp(-1/(Jρ)), ties T_K to the bandwidth D_0 and the coupling Jρ between the impurity and the electron sea.

Recognition Science models this coupling with its cost function J(x) = (x + 1/x)/2 - 1, a quantity that measures the price of a recognition event. The framework's library, a machine-checked collection of formal theorems, defines a domain cost as J(m/e), where m and e are positive real numbers. The library establishes three general facts about this cost: it vanishes when m equals e, it is never negative for positive inputs, and the threshold phi - 3/2 is positive. These are theorems about the cost function itself, not about any specific physical system.

The Kondo temperature enters as a research note, not a formal result. The note evaluates the coupling at the golden ratio phi, giving J(phi) = 0.118, and plugs this into the standard Kondo formula to estimate T_K ~ 2 K for a bandwidth of 10^4 K, consistent with typical Kondo systems. But the module defines no physical meaning for m and e in terms of the impurity or the electron band. Without that definition, the cost function remains a general mathematical object, and the Kondo temperature calculation is an unformalized suggestion.

What the module does establish is a template: a certified structure that packages the three cost theorems, with a demonstration that such a structure exists. This template is shared verbatim across 2383 sibling modules, each a placeholder for a subject-specific theory. The Kondo physics would become a theorem only when m and e are defined in the impurity system's own terms. Until then, the module proves the scaffolding, not the building.

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

The Kondo temperature formula T_K = D_0 exp(-1/(Jρ)) is not derived in this module. The numerical estimate T_K ~ 2 K is not a proved result, only a research note. The cost function J(m/e) is not shown to describe any specific impurity system.

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/Anderson_Impurity2.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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