OAI.Laughlin.planar_gap

The theorem states that, for every particle number N and every homogeneous state ψ, γ*·E(ψ) ≤ S(ψ) as extended nonnegative reals, where γ* = 4616733319001/10^14 ≈ 0.04617. A homogeneous state assigns to each L a vector ψ_L in the exterior algebra over ℂ^{L+1} (orbitals 0,…,L), whose coordinates in the occupation basis indexed by subsets A of {0,…,L} vanish unless A has exactly N elements and its indices sum to exactly L. For each p from 0 to 2L, the pair operator P_p is the matrix on occupation subsets given by Σ_{i,j} c_p(i,j) a_j a_i, where a_i annihilates orbital i and c_p(i,j) = (i−j)·√(p!/(2^p i! j!))/2 if i+j = p+1 and 0 otherwise. The energy E(ψ) is the sum over L of the real number Σ_{p=0}^{2L} ‖P_p ψ_L‖², the squared norm taken over occupation coordinates. The quantity S(ψ) is the sum over L of ‖Hψ_L‖², where H = Σ_{p=0}^{2L} P_p* P_p, with P_p* the conjugate transpose of P_p. Both infinite sums over L are taken in [0,∞], and the statement is the stated inequality for all such ψ.