Mean field andε-expansion study of spin glasses

A Landau-Ginzburg phenomenological free energy for the Edwards and Anderson spin-glass model when there is competition between spin-glass and ferromagnetic ordering is developed. This free energy obtained with the use of the $n\to 0$ replication procedure is analyzed using mean-field theory and the $\epsilon$ expansion. Critical exponents for the ferromagnetic-spin-glass multicritical point are calculated in $6-\epsilon$ dimensions. For Ising systems, $\nu =\frac{1}{2}+\left(\frac{2}{3}\right)\epsilon$ and $\phi =1+\left(\frac{1}{2}\right)\epsilon$. For $\mathrm{XY}$ and Heisenberg systems, these exponents are complex. This result is not fully understood. The Harris-Plischke-Zuckermann model for amorphous magnetism is shown to have an Ising-like spin-glass fixed point in high enough dimension.