Critical behavior near the metal-insulator transition in the one-dimensional extended Hubbard model at quarter filling

Abstract

We examine the critical behavior near the metal-insulator transition (MIT) in the one-dimensional extended Hubbard model with the on-site and the nearest-neighbor interactions $U$ and $V$ at quarter filling using a combined method of the numerical diagonalization and the renormalization group (RG). The Luttinger-liquid parameter ${K}_{\ensuremath{\rho}}$ is calculated with the exact diagonalization for finite size systems and is substituted into the RG equation as an initial condition to obtain ${K}_{\ensuremath{\rho}}$ in the infinite size system. This approach also yields the charge gap $\ensuremath{\Delta}$ in the insulating state near the MIT. The results agree very well with the available exact results for $U=\ensuremath{\infty}$ even in the critical regime of the MIT where the characteristic energy becomes exponentially small and the usual finite size scaling is not applicable. When the system approaches the MIT critical point $V\ensuremath{\rightarrow}{V}_{c}$ for a fixed $U$, ${K}_{\ensuremath{\rho}}$ and $\ensuremath{\Delta}$ behave as ${\ensuremath{\mid}\mathrm{l}\mathrm{n}\ensuremath{\Delta}\ensuremath{\mid}}^{\ensuremath{-}2}={c}_{\ensuremath{\Delta}}(V∕{V}_{c}\ensuremath{-}1)$ and ${({K}_{\ensuremath{\rho}}\ensuremath{-}1∕4)}^{2}={c}_{K}(1\ensuremath{-}V∕{V}_{c})$, where the critical value ${V}_{c}$ and the coefficients ${c}_{\ensuremath{\Delta}}$ and ${c}_{K}$ are functions of $U$. These critical properties, which are known to be exact for $U=\ensuremath{\infty}$, are observed also for finite $U$ case. We also observe the same critical behavior in the limit of the MIT critical point $U\ensuremath{\rightarrow}{U}_{c}$ when $U$ is varied for a fixed $V$.