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Experimental measurements of nonlinear dielectric response in glassformerslike supercooled glycerol or propylene carbonate have been interpreted asproviding evidence for a growing thermodynamic length scale when loweringtemperature. A heuristic picture based on coherently flipping `superdipoles'with disordered internal structure has been argued to capture the essence ofthe experimentally reported behavior, pointing to the key role of effectivelydisordered interactions in structural glasses. We test these ideas by devisingan explicit one-dimensional model of interacting spins incorporating both thespin-glass spirit of the superdipole argument, and the necessary long-timedecorrelation of structural disorder, encoded here in a slow dynamics of thecoupling constants. The frequency-dependent third-order response of the modelqualitatively reproduces the typical humped shape reported in experiments. Thetemperature dependence of the maximum value is also qualitatively reproduced.In contrast, the humped shape of the third-order response is not reproduced bya simple kinetically constrained spin model with non-interacting spins. Torationalize these results, we propose a two-length-scale scenario bydistinguishing between the characteristic length of dynamical heterogeneitiesand a coherence length that monitors the effect of interactions. We show thatboth length scales are identical in the kinetically constrained spin model,while they have significantly different dynamics in the model of interactingspins.

Nonlinear dielectric response in glasses: restoring forces and avoided spin-glass criticality