Fig. 2: Self-consistent Hartree–Fock.

a–c The renormalized band parameters \(\tilde{\epsilon },\tilde{\Delta },E\) as a function of the free energy ϵ within self-consistent Hartree–Fock. For α < 1, \(\tilde{\epsilon }\) and \(\tilde{\Delta }\) vary linearly with ϵ near ϵ = 0. The corresponding slope diverges as α → 1, and becomes non-analytic at α = 1, developing a discontinuity in the band parameters at ϵ = 0 for α > 1, resulting in a hard gap. (d) The corresponding density of states within self-consistent Hartree–Fock, very similar behavior is found with an explicit hard gap for α > 1.