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The interplay between quantum fluctuations and disorder is investigated in aspin-glass model, in the presence of a uniform transverse field $\Gamma$, and alongitudinal random field following a Gaussian distribution with width$\Delta$. The model is studied through the replica formalism. This study ismotivated by experimental investigations on the LiHo$_x$Y$_{1-x}$F$_4$compound, where the application of a transverse magnetic field yields ratherintriguing effects, particularly related to the behavior of the nonlinearmagnetic susceptibility $\chi_3$, which have led to a considerable experimentaland theoretical debate. We analyzed two situations, namely, $\Delta$ and$\Gamma$ considered as independent, as well as these two quantities related asproposed recently by some authors. In both cases, a spin-glass phase transitionis found at a temperature $T_f$; moreover, $T_f$ decreases by increasing$\Gamma$ towards a quantum critical point at zero temperature. The situationwhere $\Delta$ and $\Gamma$ are related appears to reproduce better theexperimental observations on the LiHo$_x$Y$_{1-x}$F$_4$ compound, with thetheoretical results coinciding qualitatively with measurements of the nonlinearsusceptibility. In this later case, by increasing $\Gamma$, $\chi_3$ becomesprogressively rounded, presenting a maximum at a temperature $T^*$ ($T^*>T_f$).Moreover, we also show that the random field is the main responsible for thesmearing of the nonlinear susceptibility, acting significantly inside theparamagnetic phase, leading to two regimes delimited by the temperature $T^*$,one for $T_f<T<T^*$, and another one for $T>T^*$. It is argued that theconventional paramagnetic state corresponds to $T>T^*$, whereas the temperatureregion $T_f<T<T^*$ may be characterized by a rather unusual dynamics, possiblyincluding Griffiths singularities.

Nonlinear susceptibility of a quantum spin glass under uniform transverse and random longitudinal magnetic fields