Mean-field theory andεexpansion for Anderson localization

A general field-theoretic formulation of the Anderson model for the localization of wave functions in a random potential is given in terms of $n$-component replicated fields in the limit $n\to 0$, and is analyzed primarily for spatial dimension $d\ge 4$. Lengths ${\xi }_1$ and ${\xi }_2$ associated with the spatial decay of correlations in the single-particle and two-particle Green's functions, respectively, are introduced. Two different regimes, the weak coupling and strong coupling, are distinguished depending on whether ${\xi }_1^{-1\mathrm{}}$ or ${\xi }_2^{-1\mathrm{}}$, respectively, vanishes as the mobility energy, $E_c$, is approached. The weak-coupling regime vanishes as $d\to 4+$. Mean-field theory is developed from the uniform minimum of the Lagrangian for both the strong- and weak-coupling cases. For the strong-coupling case it gives the exponents $v_a=\frac{1}{4}$, ${\gamma }_a={\beta }_a=\frac{1}{2}$, $\eta =0\mathrm{}$, and $\mu =1\mathrm{}$, where ${\beta }_a$ is the exponent associated with the density of extended states and $\mu$ is that associated with the conductivity. Simple heuristic arguments are used to verify the correctness of these unusual mean-field values. Infrared divergences in perturbation theory for the strong-coupling case occur for $d<\mathrm{}8\mathrm{}$, and an $\epsilon$ expansion ($\epsilon =8-d$) is developed which is found to be identical to that previously analyzed for the statistics of lattice animals and which gives ${\beta }_a=\frac{1}{2}-\frac{\epsilon }{12}$, $\eta =-\frac{\epsilon }{9}$, $v_a=\frac{1}{4}+\frac{\epsilon }{36}$, and $\mu =1-\frac{5\epsilon }{36}$. The results are consistent with the Ward identity, which in combination with scaling arguments requires that ${\beta }_a+{\gamma }_a=1\mathrm{}$. The treatment takes account of the fact that the average of the on-site Green's function ${\left[G\right(\overrightarrow{\mathrm{x}},\overrightarrow{\mathrm{x}};E\left)\right]}_{\mathrm{av}}$ is nonzero and is predicated on this quantity being real, i.e., on the density of states vanishing at the mobility edge. We also show that localized states emerge naturally from local minima of finite action in the Lagrangian. These instanton solutions are analyzed on a lattice where the cutoff produced by the lattice constant leads to lattice...