Prove2Me

One-body density matrix of a Slater determinant is the occupied projector (Eq. (43))

Proved
KochHF.slater_one_body_density

by Lucas · Oct 9, 2026 · Mathlib 0df444a (Lean v4.33.1)

hartree-fockmathematical-physicssecond-quantization

Let α1,…,αN∈CK\alpha_1,\dots,\alpha_N\in\mathbb C^K be orthonormal, ∣Φ⟩=cαN†⋯cα1†∣0⟩|\Phi\rangle=c^\dagger_{\alpha_N}\cdots c^\dagger_{\alpha_1}|0\rangle and P=∑n∣αn⟩⟨αn∣P=\sum_n|\alpha_n\rangle\langle\alpha_n|. For all reference orbitals ϕn,ϕm\phi_n,\phi_m,

Γnm(1)=⟨Φ∣cn†cm∣Φ⟩=⟨ϕm∣P∣ϕn⟩.\Gamma^{(1)}_{nm}=\langle\Phi|c^\dagger_n c_m|\Phi\rangle=\langle\phi_m|P|\phi_n\rangle .

As an operator on the one-electron space the one-body density matrix of a Slater determinant is therefore the projector onto the occupied subspace.

Preamble
import Mathlib
import Definitions.Def_KochHF_FockSpace

open Matrix
Formal statement
namespace KochHF

theorem slater_one_body_density {K N : ℕ} (α : Fin N → Fin K → ℂ)
    (hα : ∀ i j, orbInner (α i) (α j) = if i = j then 1 else 0) (n m : Fin K) :
    matEl (slater α) (cdag n * cann m) (slater α) =
      orbInner (Pi.single m 1) (occProj α (Pi.single n 1)) := by sorry

end KochHF
Source
E. Koch, *Mean-Field Theory: Hartree-Fock and BCS*, in E. Pavarini, E. Koch, J. van den Brink, G. Sawatzky (eds.), Quantum Materials: Experiments and Theory, Modeling and Simulation Vol. 6, Forschungszentrum Jülich, 2016, ISBN 978-3-95806-159-0, http://www.cond-mat.de/events/correl16, Sec. 2.4, p. 2.12, Eq. (43).
Read-back

What the Lean code literally says, in plain math · Aristotle (Harmonic) — NON-BLIND: same agent that drafted the statements

Non-blind read-back — not independent testimony. This read-back was written by the same agent that drafted the Lean statement, with full knowledge of the source text and of the intended meaning. It is not a blind audit and must not be mistaken for independent testimony; reviewers should compare the Lean code against the source themselves.

Throughout, K∈NK\in\mathbb N is the number of orthonormal reference orbitals ϕ0,…,ϕK−1\phi_0,\dots,\phi_{K-1}; an orbital is a coefficient vector φ∈CK\varphi\in\mathbb C^K and ⟨φ∣χ⟩=∑μφμ‾χμ\langle\varphi|\chi\rangle=\sum_\mu\overline{\varphi_\mu}\chi_\mu. Fock space is C2K\mathbb C^{2^K} with orthonormal basis ∣S⟩|S\rangle, S⊆{0,…,K−1}S\subseteq\{0,\dots,K-1\}; ci†∣S⟩=(−1)#{j∈S: j>i}∣S∪{i}⟩c^\dagger_i|S\rangle=(-1)^{\#\{j\in S:\,j>i\}}|S\cup\{i\}\rangle for i∉Si\notin S and 00 for i∈Si\in S; ci=(ci†)†c_i=(c_i^\dagger)^\dagger (conjugate transpose); cφ†=∑μφμcμ†c^\dagger_\varphi=\sum_\mu\varphi_\mu c^\dagger_\mu, cφ=∑μφμ‾cμc_\varphi=\sum_\mu\overline{\varphi_\mu}c_\mu; ∣0⟩=∣∅⟩|0\rangle=|\emptyset\rangle; for orbitals α0,…,αN−1\alpha_0,\dots,\alpha_{N-1} the vector ∣Φα⟩=cαN−1†⋯cα0†∣0⟩|\Phi_\alpha\rangle=c^\dagger_{\alpha_{N-1}}\cdots c^\dagger_{\alpha_0}|0\rangle (for N=0N=0 this is ∣0⟩|0\rangle). For all K,NK,N, every orthonormal family α0,…,αN−1∈CK\alpha_0,\dots,\alpha_{N-1}\in\mathbb C^K (hypothesis ⟨αi∣αj⟩=δij\langle\alpha_i|\alpha_j\rangle=\delta_{ij}) and all reference indices n,m∈{0,…,K−1}n,m\in\{0,\dots,K-1\}:

⟨Φα∣ cn†cm ∣Φα⟩=⟨em ∣ Pen⟩=∑kαk,m αk,n‾,\langle\Phi_\alpha|\,c^\dagger_n c_m\,|\Phi_\alpha\rangle=\langle e_m\,|\,P e_n\rangle=\sum_{k}\alpha_{k,m}\,\overline{\alpha_{k,n}},

where en∈CKe_n\in\mathbb C^K is the nn-th standard unit vector (the reference orbital ϕn\phi_n) and Pen=∑k⟨αk∣en⟩αkPe_n=\sum_k\langle\alpha_k|e_n\rangle\alpha_k. Here cn†,cmc^\dagger_n,c_m are the reference-orbital operators.

Human review

  • Endorsed by Shuze Chen · Oct 10, 2026

    Confirmed by the moderator at approval.

  • Endorsed by Lucas · Oct 10, 2026

    Confirmed by the mission captain (proposal self-audit).

View graph