Nonlinear susceptibility in glassy systems: A probe for cooperative dynamical length scales

We argue that for generic systems close to a critical point, an extended fluctuation-dissipation relation connects the low frequency nonlinear (cubic) susceptibility to the four-point correlation function. In glassy systems, the latter contains interesting information on the heterogeneity and cooperativity of the dynamics. Our result suggests that if the abrupt slowing down of glassy materials is indeed accompanied by the growth of a cooperative length $\ensuremath{\ell}$, then the nonlinear, $3\ensuremath{\omega}$ response to an oscillating field (at frequency $\ensuremath{\omega}$) should substantially increase and give direct information on the temperature (or density) dependence of $\ensuremath{\ell}$. The analysis of the nonlinear compressibility or the dielectric susceptibility in supercooled liquids, or the nonlinear magnetic susceptibility in spin-glasses, should give access to a cooperative length scale, that grows as the temperature is decreased or as the age of the system increases. Our theoretical analysis holds exactly within the mode-coupling theory of glasses.