Fluctuations in Short-Range Spin-Glasses

Abstract

Spatial fluctuations in spin-glasses are studied by an expansion to quadratic order around the dynamic mean-field theory. Below $T_c$, a spatial decay of the "spin-glass susceptibility" ${\left[{\chi }^2\right(\overrightarrow{\mathrm{R}}, \omega \left)\right]}_{\mathrm{av}}$ is characterized at infinitesimal frequencies by a correlation length $\xi \left(\omega \right)$ which diverges as the inverse of the time-dependent Edwards-Anderson order parameter $q\left(\frac{1}{\omega }\right)$ in the limit $\omega \to 0$. The Fourier transform of equal-time correlations ${\left[{〈S\left(\overrightarrow{0}\right)S\left(\overrightarrow{\mathrm{R}}\right)〉}^2\right]}_{\mathrm{av}}$ diverges as $k^{-4\mathrm{}}$, signaling a breakdown of mean-field theory at low $T$ below $d=4\mathrm{}$.