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
Vc and the coefficients ${c}_{\ensuremath{\Delta}}$ and
cK 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.
The concept of F invariance, which previously arose in our analysis of the integral and half-integral quantum Hall effects, is studied in $2+2\ensuremath{\varepsilon}$ spatial dimensions. We report the results of a detailed renormalization-group analysis, and establish the renormalizability of the (Finkelstein) action to two-loop order. We show that the infrared behavior of the theory can be extracted from gauge-invariant (F-invariant) quantities only. For these quantities (conductivity, specific heat) we derive explicit scaling functions. We identify a bosonic quasiparticle density of states which develops a Coulomb gap as one approaches the metal-insulator transition from the metallic side. We discuss the consequences of F invariance for the strong-coupling, insulating regime.
We consider the Finkelstein action describing a system of spin-polarized or spinless electrons in $2+2\ensuremath{\epsilon}$ dimensions, in the presence of disorder as well as the Coulomb interactions. We extend the renormalization-group analysis of our previous work and evaluate the metal-insulator transition of the electron gas to second order in an $\ensuremath{\epsilon}$ expansion. We obtain the complete scaling behavior of physical observables like the conductivity and the specific heat with varying frequency, temperature, and/or electron density. We extend the results for the interacting electron gas in $2+2\ensuremath{\epsilon}$ dimensions to include the quantum critical behavior of the plateau transitions in the quantum Hall regime. Although these transitions have a very different microscopic origin and are controlled by a topological term in the action $(\ensuremath{\theta}$ term), the quantum critical behavior is in many ways the same in both cases. We show that the two independent critical exponents of the quantum Hall plateau transitions, previously denoted as $\ensuremath{\nu}$ and p, control not only the scaling behavior of the conductances ${\ensuremath{\sigma}}_{\mathrm{xx}}$ and ${\ensuremath{\sigma}}_{\mathrm{xy}}$ at finite temperatures T, but also the non-Fermi-liquid behavior of the specific heat ${(c}_{v}\ensuremath{\propto}{T}^{p}).$ To extract the numerical values of $\ensuremath{\nu}$ and p it is necessary to extend the experiments on transport to include the specific heat of the electron gas.