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 $d_c^0<d<\mathrm{}{d}_c$ at low temperatures $T$, the magnetization has a discontinuity (from zero to $M$) when $\Delta$ (the average square field) approaches zero. The correlation length $\xi$ diverges as ${\Delta }^{{-\nu }_{\Delta }}$. The structure factor is shown to scale as $S(q,\xi )={\xi }^d\overline{S}\left(q\xi \right)$. Simple assumptions on scaling near $T=0\mathrm{}$ yield ${\nu }_{\Delta }=\frac{1}{(d_c-d)}$, with $d_c=4\mathrm{}$ for continuous symmetry spins and $d_c=2\mathrm{}$ for Ising spins.