Application of the renormalization group to phase transitions in disordered systems

Abstract

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