OAI.LaughlinGap.thm_main

The theorem states that there is a threshold N₀ ≥ 2 such that for every N ≥ N₀ and every antisymmetric complex-valued state ψ on N particles, each with local levels 0,…,Q where Q = 3(N−1), one has (1/25)·d(ψ)² ≤ E(ψ). Here a state assigns a complex number to each configuration a : Fin N → {0,…,Q}, and antisymmetric means that swapping the values at two distinct positions i and j negates ψ. The energy E(ψ) sums, over pairs i<j, over p = 0,…,2Q−2, and over configurations a with a_i = a_j = 0, the squared modulus of the pair amplitude, which is the sum over x,y of pairCoefficient(Q,p,x,y)·ψ(a with a_i replaced by x and a_j by y). The pair coefficient vanishes unless x+y = p+1, in which case it equals (x−y)/√2 times the square root of Q^{(x)}·Q^{(y)}·p! divided by Q·(2Q−2)^{(p)}·x!·y!, where m^{(k)} is the descending factorial. The Laughlin vector is built from the polynomial ∏{i<j}(x{i,0}x_{j,1} − x_{j,0}x_{i,1})³ in variables indexed by particle and a Boolean: its coefficient at the monomial with exponent a_i on x_{i,1} and Q−a_i on x_{i,0} is divided by ∏_i √C(Q,a_i). The quantity d(ψ)² is the infimum over complex c of the sum over configurations of |ψ(a) − c·Laughlin(a)|², the squared distance from ψ to the line spanned by the Laughlin vector, with no normalization of ψ assumed.