Identification of the free-fermion second-order projection

Establish whether the second-order inter-channel spectral projection of the exact free-fermion covariance, denoted by \mathcal G^{(2)}_{\rm FF}, coincides with the mean-sector DCA level g^{(2)} under an appropriate mapping.

Background

The paper distinguishes the exactly computed two-channel ridge weight \bar C_i from the hierarchy’s mean-sector level g^{(2)}. The former is a projection at coincident eigenstates, whereas the latter is defined through interaction-vertex structure in the DCA expansion.

Because the random free-fermion Hamiltonian is quadratic and has no many-body interaction vertices in the hierarchy’s V-counting, the authors do not identify these quantities. The open problem is to determine whether an appropriate channel-multiplicity or other mapping can make the exact covariance projection coincide with g^{(2)}.