Charge Gap in the One-Dimensional Extended Hubbard Model at Quarter Filling

We propose a new combined approach of the exact diagonalization, the renormalization group and the Bethe ansatz for precise estimates of the charge gap $\Delta$ in the one-dimensional extended Hubbard model with the onsite and the nearest-neighbor interactions $U$ and $V$ at quarter filling. This approach enables us to obtain the absolute value of $\Delta$ including the prefactor without ambiguity even in the critical regime of the metal-insulator transition (MIT) where $\Delta$ is exponentially small, beyond usual renormalization group methods and/or finite size scaling approaches. The detailed results of $\Delta$ down to of order of $10^{-10}$ near the MIT are shown as contour lines on the $U$-$V$ plane.