Critical properties of random-spin models from theεexpansion

In quenched random-spin systems, the renormalization group can be used to develop recursion relations for the probability distribution for random potentials. Alternatively, recursion relations for the average values of potentials and their higher cumulants can be obtained. In this paper, the above technique is used to study phase transitions in quenched random $n$-component classical spin systems using the $\epsilon$ expansion to second order in $\epsilon$. It there are long-range correlations in the random potentials (e.g., all potentials along a line are equal), there are no stable physical fixed points within the $\epsilon$ expansion. This is interpreted as a smeared transition. If there are no long-range correlations in the random potentials, there is a sharp transition with pure system exponents if the specific-heat exponent ${\alpha }^H$ of the pure system is negative. If ${\alpha }^H>0\mathrm{}$ and $n>\mathrm{}1\mathrm{}$, there is a sharp transition with new exponents $\eta =\left[\frac{(5\mathrm{}{n}^2-8n)}{256{\mathrm{}(n-1\mathrm{})\mathrm{}}^2}\right]{\epsilon }^2$ and $\nu =\frac{1}{2}+\left[\frac{3n}{32(n-1\mathrm{})\mathrm{}}\right]\epsilon +\left[\frac{n(127\mathrm{}{n}^2-572n-32)}{4096{\mathrm{}(n-1\mathrm{})\mathrm{}}^3}\right]{\epsilon }^2$. For $n=1\mathrm{}$, there is no stable fixed point, which is again interpreted as a smeared transition.