Generalized Schrieffer-Wolff transformation of multiflavor Hubbard models

We give a self-contained derivation of the low-energy effective interactions of the $\mathrm{SU}\left(N\right)$ Hubbard model, a multiflavor generalization of the one-band Hubbard model, by using a generalized Schrieffer-Wolff transformation (SWT). The effective interaction of doublons and holons, which has been largely ignored in previous SWT studies (e.g., the $t\text{−}J$ model), leads to distinct peaks in the local density of states. As shown by Lee et al. [Phys. Rev. Lett. 119, 236402 (2017)], this underlying effective doublon-holon interaction explains the numerical observation of the subpeaks at the inner edges of the Hubbard bands in the metallic phase close to the Mott transition.

Figure 1

Mean-field analysis of the $\text{SU}\left(N=2\right)$ Hubbard model. (a), (b) The correlation functions of doublons and holons, $A_{d{d}^{†}}\left(\omega \right)$ and $A_{dh}\left(\omega \right)$, have peaks at $\omega =\pm {\omega }_{dh}$. Gray shading at higher (lower) energy windows indicate that large (small) energy scales have been neglected by the specific type of mean-field and decoupling approximations employed on top of the generalized SWT. (c), (d) The eigenvalues ${\lambda }_k$ of $H_{\mathbf{p}}$ in Eq. (33b). Dashed vertical lines indicate the locations of ${\lambda }_k=\pm {\omega }_{dh}$. Each row of panels are for the same value of $U$. Here we used a fixed value of ${\mathrm{\Delta }}_{dh}=2.91D$, which is equal to the critical interaction strength $U_{c2}$, and the $U$-dependent values of $\langle P_{i1}\rangle$ taken from the DMFT calculations in Ref. [4].