Nonlinear response functions in an exponential trap model

The nonlinear response to an oscillating field is calculated for a kinetic trap model with an exponential density of states above the glass transition temperature T0. For temperatures not too close to T0, the results are similar to those obtained for the model with a Gaussian density of states. The choice of the dynamical variable that couples to the field and in particular its dependence on the trap energies generally has a strong impact on the shape of the dynamic response. The modulus of the frequency dependent third-order response either shows a peak or exhibits a monotonous decay from a finite low-frequency limit to a vanishing response at high frequencies depending on the dynamical variable. If a peak is observed, its height can show different temperature dependencies with the common feature of a scaling behavior near T0. Additionally, in some but not all cases the static nonlinear susceptibility diverges at T0. A recently proposed approximation that relates the cubic response to a four-time correlation function does not give reliable results due to a wrong estimate of the low-frequency limit of the response.