Flat-band energy filtering in interacting systems: conditions for improving thermoelectric performances

Motivated by recent theoretical and experimental studies on the role of flatbands in the thermoelectric properties of Ni$_3$In$_{1-x}$Sn$_x$ compounds, we investigate electron transport in two minimal one-dimensional flatband models, the sawtooth and diamond chains, which differ in a crucial aspect: the flatband is separated from the dispersive band by a finite gap in the former, while the two bands touch in the latter. Using a non-equilibrium Green function framework with interactions treated at the Hartree-Fock and GW levels, we compute the full set of thermoelectric coefficients and the figure of merit $zT$ as functions of gate voltage and temperature. We show that, contrary to naive expectation, a perfectly isolated flat-band is a physically ill-founded thermoelectric: the electrical conductivity vanishes as the chemical potential enters the flat-band, rendering the large Seebeck coefficient and the apparent violation of the Wiedemann-Franz law physically meaningless. Optimal thermoelectric performance is instead achieved just below the flat-band edge, where the transmission function varies most rapidly with energy, consistent with the Mahan-Sofo picture, and requires a finite broadening of the flat-band through hybridization with dispersive states. We further show that electron-electron interactions renormalize the flat-band structure itself, inducing an interaction-driven narrowing of the bandwidth and, in the diamond chain, a correlation-induced opening of a gap between the flat-band and the dispersive band near half-filling. Mean-field treatments are found to systematically overestimate $zT$, highlighting the importance of beyond-mean-field correlations for quantitatively reliable predictions in flat-band thermoelectrics.