TheoremStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Two-band Chern number and degree formula

Statement

Assume Axiom of Choice and Countable Choice. Let a two-orbital finite-range Bloch Hamiltonian on the oriented torus $T_{2 π}^{2}$ have the form

$h (k) = d_{0} (k) I + d_{1} (k) σ_{1} + d_{2} (k) σ_{2} + d_{3} (k) σ_{3},$

where $d_{0}, d_{1}, d_{2}, d_{3}$ are real smooth periodic functions, $σ_{1} = (\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array})$ , $σ_{2} = (\begin{array}{cc} 0 & - i \\ i & 0 \end{array})$ , $σ_{3} = (\begin{array}{cc} 1 & 0 \\ 0 & - 1 \end{array})$ are the Pauli matrices, and $d = (d_{1}, d_{2}, d_{3})$ is nowhere zero. Define $n = d / ∣ d ∣$ . The upper and lower band projectors are $P_{+} = (I + n \cdot σ) / 2$ and $P_{-} = (I - n \cdot σ) / 2$ . With the Chern convention $C (P) = (i / 2 π) \int_{T^{2}} tr (P (d P \land d P) P)$ and orientation $d k_{1} \land d k_{2}$ ,

$C (P_{+}) = - \deg (n) = - \frac{1}{4 π} \int_{T^{2}} n \cdot (\partial_{k_{1}} n \times \partial_{k_{2}} n) d k_{1} d k_{2}, C (P_{-}) = + \deg (n) = + \frac{1}{4 π} \int_{T^{2}} n \cdot (\partial_{k_{1}} n \times \partial_{k_{2}} n) d k_{1} d k_{2} .$

Statement references

Gapped band curvature and Chern invariance

Normalized area integral equals the degree

Facts & Assumptions

Given: Axiom of Choice, Countable Choice, the stated finite-range Bloch Hamiltonian, and $∣ d (k) ∣ > 0$ on the torus.

[A1]

Axiom of Choice and Countable Choice are the principles in The Axiom of Choice and The Axiom of Countable Choice ( ${A C}_{ω}$ ).

[F1]

Lattice Bloch transform and band projectors fixes the finite-range Bloch convention and torus orientation; Exact Bloch transform and regularity of gapped band projectors gives smooth Hermitian fibers and smooth isolated-band projectors.

[F2]

Gapped band curvature and Chern invariance gives the Chern integral convention and identifies it with an integer for a smooth gapped projector.

[F3]

Normalized area integral equals the degree identifies the normalized sphere-area pullback integral with the oriented homological degree.

Proof

Given: [A1], [F1]–[F3]. Write $s \in {+ 1, - 1}$ for the upper or lower band sign.

1.1A1F1givenalgebra

Direct multiplication of the displayed Pauli matrices gives $σ_{a} σ_{b} = δ_{a b} I + i \sum_{c} ϵ_{a b c} σ_{c}$ , $tr σ_{a} = 0$ , and $tr (σ_{a} σ_{b}) = 2 δ_{a b}$ . Thus $(n \cdot σ)^{2} = ∣ n ∣^{2} I = I$ , and $h = d_{0} I + ∣ d ∣ (n \cdot σ)$ has eigenvalues $d_{0} \pm ∣ d ∣$ with the stated orthogonal projectors. Since $∣ d ∣$ is continuous and positive on compact $T^{2}$ , its minimum is positive, so the two bands are uniformly separated; $n$ and $P_{s}$ are smooth.

2.1F2step 1.1givenalgebra

Differentiating $P_{s} = (I + s n \cdot σ) / 2$ gives $\partial_{j} P_{s} = (s / 2) (\partial_{j} n) \cdot σ$ . The Pauli product identity yields $[a \cdot σ, b \cdot σ] = 2 i (a \times b) \cdot σ$ , hence $[\partial_{1} P_{s}, \partial_{2} P_{s}] = \frac{i}{2} (\partial_{1} n \times \partial_{2} n) \cdot σ$ . Using $tr (P_{s} σ_{a}) = s n_{a}$ , which follows from the trace identities in step 1.1, we obtain $tr (P_{s} [\partial_{1} P_{s}, \partial_{2} P_{s}]) = \frac{i s}{2} n \cdot (\partial_{1} n \times \partial_{2} n)$ . In dimension two this is the coefficient of $tr (P_{s} (d P_{s} \land d P_{s}) P_{s})$ : cyclicity and $P_{s}^{2} = P_{s}$ give $tr (P_{s} X P_{s}) = tr (P_{s} X)$ . Multiplying by $i / (2 π)$ therefore gives $C (P_{s}) = - s (4 π)^{- 1} \int n \cdot (\partial_{1} n \times \partial_{2} n) d k_{1} d k_{2}$ .

3.1F3step 2.1algebra∎

The normalized area form in [F3] pulls back to $(4 π)^{- 1} n \cdot (\partial_{1} n \times \partial_{2} n) d k_{1} \land d k_{2}$ , by the defining outward area form on the unit sphere. Hence its integral is $\deg (n)$ by [F3]. Substitution into step 2.1 gives $C (P_{s}) = - s \deg (n)$ , which is the two displayed upper- and lower-band formulas.

Remarks

The cited Qi–Wu–Zhang treatment of Asbøth, Oroszlány, and Pályi uses the same two-band lattice Hamiltonian and gap-closing values, but defines its band invariant with the opposite sign: its Eqs. (2.40)–(2.41) give $Q = - (2 π)^{- 1} \int (\partial_{k_{x}} A_{y} - \partial_{k_{y}} A_{x})$ for $A_{j} = i ⟨ u ∣ \partial_{k_{j}} u ⟩$ . The $C (P)$ convention here instead gives $+ (2 π)^{- 1} \int d A$ . Accordingly, the signs in this theorem and its examples come from the explicit projector and Pauli calculation above; the cited table is not copied without conversion.

Two-band Chern number and degree formula

Statement

Statement references

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources