Sufficient Conditions for a Josephson Inequality

Abstract

An inequality $d\nu f\ge 2-\alpha$ is derived under conditions that are known to hold rigorously for Ising ferromagnets. Here $d$ is dimensionality and $\nu$ and $\alpha$ are critical exponents expressed in customary notation, while $f$ is a factor that is shown to be unity when $1\ge \alpha \ge {\alpha }^{\prime }\ge 0$, and in this case the inequality reduces to one proposed by Josephson. For $0<\mathrm{}\alpha <{\alpha }^{\prime }<1<\delta$, $f$ is bounded from above to yield, in combination with the first result, $d\nu \ge (2-\alpha ) min\left(1,1-\frac{2({\alpha }^{\prime }-\alpha )}{(\delta -1)(2-\alpha )(1-{\alpha }^{\prime })}\right).$