We determine the $\beta$ function of the $\mathrm{O}\mathrm{}\left(n\right)\mathrm{}$ nonlinear $\sigma$ model in $2+\epsilon$ dimensions to four-loop order using the recently calculated four-loop-order $\zeta$ function and the critical exponent $\eta$ through order $\frac{1}{n^3}$. This $\beta$ function determines completely, according to Hikami, the four-loop-order $\beta$ function for a large class of nonlinear $\sigma$ models. As an application the conductivity exponent of the Anderson metal-insulator transition is calculated for the unitary case to order $\epsilon$. This exponent turns out to be smaller than ½.

• Received 21 July 1986

DOI:https://doi.org/10.1103/PhysRevLett.57.1383