Application of the renormalization group to phase transitions in disordered systems

The critical behavior of spin systems with quenched disorder is studied by renormalization-group methods. For the randomly dilute m-vector model, the $n=0$ limit is used to construct a translationally invariant effective Hamiltonian which describes the original disordered system. This Hamiltonian is analyzed in the ε expansion to order ε². Sharp second-order phase transitions with exponents which do not depend continuously on impurity concentration are predicted. For m>mc≡4-4ε+O(ε²) the isotropic m-component fixed point, which characterizes the critical behavior of the pure system, is stable. For m<mc, a new random fixed point becomes stable. The exponents corresponding to this fixed point are η=[(5m²-8m)256(m-1)²]ε²+O(ε³), ${ν}=1/2+[{3m}{32(m{-}1)}]{ε}+[{m(127{m}²{-}572m{-}32)}{4096{(m{-}1)}³}]{{ε}}²+O({{ε}}³)$ for $m{≠}1$, and ${η}={-}{{ε}}{106}+O({{ε}}3/2)$, ${ν}=1/2+{{({6{ε}}{53})}1/2}{4}+O({ε})$ for $m=1$. More general random systems are qualitatively discussed from the effective-Hamiltonian viewpoint.