Two-level correlation function of critical random-matrix ensembles

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The two-level correlation function Rd,ᵦ(s) of d-dimensional disordered models (d=1, 2, and 3) with long-range random-hopping amplitudes is investigated numerically at criticality. We focus on models with orthogonal (β=1) or unitary (β=2) symmetry in the strong (bᵈ ≪ 1) coupling regime, where the parameter b⁻ᵈ plays the role of the coupling constant of the model. It is found that Rd,ᵦ(s) is of the form Rd,ᵦ(s)=1+δ(s)-Fβ(sβ/bᵈβ), where F₁(x)=erfc(ad,ᵦ x) and F₂(x)= (-ad,ᵦ x²), with ad,ᵦ being a numerical coefficient depending on the dimensionality and the universality class. Finally, the level number variance and the spectral compressibility are also considerded.