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Experimental measurements of nonlinear dielectric response in glass formers like supercooled glycerol or propylene carbonate have been interpreted as providing evidence for a growing thermodynamic length scale when lowering temperature. A heuristic picture based on coherently flipping "superdipoles" with disordered internal structure has been argued to capture the essence of the experimentally reported behavior, pointing to the key role of effectively disordered interactions in structural glasses. We test these ideas by devising an explicit one-dimensional model of interacting spins incorporating both the spin-glass spirit of the superdipole argument and the necessary long-time decorrelation of structural disorder, encoded here in a slow dynamics of the coupling constants. The frequency-dependent third-order response of the model qualitatively reproduces the typical humped shape reported in experiments. The temperature dependence of the maximum value is also qualitatively reproduced. In contrast, the humped shape of the third-order response is not reproduced by a simple kinetically constrained spin model with noninteracting spins. To rationalize these results, we propose a two-length-scale scenario by distinguishing between the characteristic length of dynamical heterogeneities and a rigidity length that accounts for the local tendency of spins to flip coherently as a block, in the presence of interactions. We show that both length scales are identical in the kinetically constrained spin model, while they have significantly different dynamics in the model of interacting spins.

We studied the physical aging of glycerol in response to upward temperature steps of amplitude ranging from 0.3 to 18 K. This was done using a specially designed experimental setup allowing quick heating of a liquid film while measuring the evolution of its dielectric properties. Despite the nonlinear evolution of these observables for large steps, a fictive temperature could be obtained. In the case of moderate step amplitudes, we checked that the material time approach in its simplest form, the single parameter aging (SPA), applies well. The memory kernel extracted from the quasi-linear regime was used to test its frontiers of validity for significant step amplitudes. We showed that the observations deviate from the prediction of the material time framework and of SPA simultaneously. As these approaches link aging to equilibrium dynamics, our results help set the bounds beyond which new theoretical arguments are needed.