Low-temperature scaling for systems with random fields and anisotropies

Abstract

Random fields (or anisotropies) shift the lower critical dimensionality of spin systems from ${d}_{c}^{0}$ to ${d}_{c}$. For dimensionalities ${d}_{c}^{0}<d<{d}_{c}$ at low temperatures $T$, the magnetization has a discontinuity (from zero to $M$) when $\ensuremath{\Delta}$ (the average square field) approaches zero. The correlation length $\ensuremath{\xi}$ diverges as ${\ensuremath{\Delta}}^{{\ensuremath{-}\ensuremath{\nu}}_{\ensuremath{\Delta}}}$. The structure factor is shown to scale as $S(q,\ensuremath{\xi})={\ensuremath{\xi}}^{d}\overline{S}(q\ensuremath{\xi})$. Simple assumptions on scaling near $T=0$ yield ${\ensuremath{\nu}}_{\ensuremath{\Delta}}=\frac{1}{({d}_{c}\ensuremath{-}d)}$, with ${d}_{c}=4$ for continuous symmetry spins and ${d}_{c}=2$ for Ising spins.