OAI.LaughlinFock.thm_fock

The theorem states that for every real γ with 0 < γ < gammaStar, where gammaStar = 4616733319001/10^14 ≈ 0.04617, there exists an integer Qγ ≥ 1 such that for every integer Q ≥ Qγ the Fock inequality holds at level Q (with Q converted to a natural number) and parameter γ. Here the Fock space has basis the subsets of the Q+1 orbitals Fin(Q+1), so operators are complex matrices indexed by such occupation sets. The annihilator of orbital j sends an occupation set B containing j to B with j removed, with sign (−1) raised to the number of occupied orbitals below j, and is zero otherwise. For each p, the pair annihilator is the sum over orbitals i < j of the real coefficient √2 times the base pair coefficient, times the product of annihilator j and annihilator i. The base coefficient is zero unless i + j = p + 1, in which case it equals (i − j) times the square root of [Q!/(Q−i)! · Q!/(Q−j)! · p!] / [Q · (2Q−2)!/(2Q−2−p)! · i! · j!], divided by √2. The Hamiltonian H is the sum over p from 0 to 2Q−2 of the conjugate transpose of the pair annihilator times the pair annihilator. FockInequality(Q, γ) is the statement that the matrix H² − γH is positive semidefinite.