Fermionic eigenvector moment flow

We exhibit new functions of the eigenvectors of the Dyson Brownian motion which follow an equation similar to the Bourgade-Yau eigenvector moment flow. These observables can be seen as a Fermionic counterpart to the original (Bosonic) ones. By analyzing both Fermionic and Bosonic observables, we obtain new correlations between eigenvectors. The fluctuations $\sum_{\alpha\in I}u_k(\alpha)^2-{\vert I\vert}/{N}$ decorrelate for distinct eigenvectors as the dimension $N$ grows and an optimal estimate on the partial inner product ${\sum }_{\alpha \in I}{u}_k(\alpha )u_{\mathrm{\ell }}(\alpha )$ between two eigenvectors is given. These static results obtained by integrable dynamics are stated for generalized Wigner matrices and should apply to wide classes of mean field models.