Second-quantized theory of Anderson localization ind=2+ε

Abstract

Renormalized perturbation-theory methods with dimensional regularization are applied to the localization transition of electrons in random potentials within a field theory where the generating functional is the configurational averaged ${〈Z^n〉}_{\mathrm{av}}$ for $Z$ the vacuum-to-vacuum amplitude expressed as a functional integral over Grassman fields, in the replica limit $n=0\mathrm{}$. The bare parameters of the theory are the Fermi level $E_0$ and the variance of the random potential $W_0$. Power counting says that at dimensionality $d=2+\epsilon$ the theory is super-renormalizable. Renormalization of the inverse propagator by the definition of renormalized Fermi level and interaction, $E_F$ and $u=\left(\frac{\pi W_0}{E_F}\right){\kappa }^{d-2\mathrm{}}$, respectively, in order to cancel the single dimensional pole, leads to the same Wilson function $\beta \mathrm{}\left(u\right)\mathrm{}$ as for the compact nonlinear $\sigma$ model when the scale parameter $\kappa$ is varied at constant $\frac{W_0}{E_0}$. The conductivity is calculated in a perturbation expansion in ${\left(\tau E_F\right)}^{-1\mathrm{}}$, with ${\tau }^{-1\mathrm{}}=\pi W_0{E}_F^{\frac{d}{2}-1\mathrm{}}$ being the inverse lifetime, at $d=2+\epsilon$. It is explicitly shown that in ultraviolet-divergent $d$-dimensional loop integrals over advanced and retarded propagators, the leading term is regular while the dimensional pole occurs in the next-to-leading term. Then to leading order the conductivity ${\sigma }_0\left(\omega \right)$ is regular while the first correction that includes the diffusion modes has a dimensional pole with residue $\approx {\left(\frac{i\omega }{{\kappa }^2}\right)}^{\frac{\epsilon }{2}}$. To cancel this pole a renormalized inverse conductance $t\left(u\right)\mathrm{}$ is defined, and the new Wilson function $\beta \mathrm{}\left(t\right)\mathrm{}$ obtained by varying $\kappa$ at constant "bare" interaction $\frac{W_0}{E_F}$ coincides with the scaling theory of Abrahams, Anderson, Licciardello, and Ramakrishnan. Scaling laws are derived from the solution of the renormalization-group equation for the conductivity.