Superposition principle and nonlinear response in spin glasses

The extended principle of superposition has been a touchstone of spin-glass dynamics for almost 30 years. The Uppsala group has demonstrated its validity for the metallic spin glass, CuMn, for magnetic fields H up to 10 Oe at the reduced temperature Tr=T/Tg=0.95, where Tg is the spin-glass condensation temperature. For H>10 Oe, they observe a departure from linear response which they ascribe to the development of nonlinear dynamics. The thrust of this paper is to develop a microscopic origin for this behavior by focusing on the time development of the spin-glass correlation length, ζ(t,tw;H). Here, t is the time after H changes, and tw is the time from the quench for T>Tg to the working temperature T until H changes. We connect the growth of ζ(t,tw;H) to the barrier heights Δ(tw) that set the dynamics. The effect of H on the magnitude of Δ(tw) is responsible for affecting differently the two dynamical protocols associated with turning H off (TRM, or thermoremanent magnetization) or on (ZFC, or zero-field-cooled magnetization). This difference is a consequence of nonlinearity based on the effect of H on Δ(tw). Superposition is preserved if Δ(tw) is linear in the Hamming distance Hd (proportional to the difference between the self-overlap qEA and the overlap q[Δ(tw)]). However, superposition is violated if Δ(tw) increases faster than linear in Hd. We have previously shown, through experiment and simulation, that the barriers Δ(tw) do increase more rapidly than linearly with Hd through the observation that the growth of ζ(t,tw;H) slows down as ζ(t,tw;H) increases. In this paper, we display the difference between the zero-field-cooled ζZFC(t,tw;H) and the thermoremanent magnetization ζTRM(t,tw;H) correlation lengths as H increases, both experimentally and through numerical simulations, corresponding to the violation of the extended principle of superposition in line with the finding of the Uppsala Group.