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

The interplay between quantum fluctuations and disorder is investigated in a quantum spin-glass model, in the presence of a uniform transverse field , as well as of a longitudinal random field hi , which follows a Gaussian distribution characterized by a width proportional to . The interactions are infinite-ranged, and the model is studied through the replica formalism, within a one-step replica-symmetry-breaking procedure; in addition, the dependence of the Almeida-Thouless eigenvalue λAT (replicon) on the applied fields is analyzed. This study is motivated by experimental investigations on the LiHoxY1−xF4 compound, where the application of a transverse magnetic field yields rather intriguing effects, particularly related to the behavior of the nonlinear magnetic susceptibility χ3, which have led to a considerable experimental and theoretical debate. We have analyzed two physically distinct situations, namely, and considered as independent, as well as these two quantities related, as proposed recently by some authors. In both cases, a spin-glass phase transition is found at a temperature Tf , with such phase being characterized by a nontrivial ergodicity breaking; moreover, Tf decreases by increasing towards a quantum critical point at zero temperature. The situationwhere and are related [ ≡ ( )] appears to reproduce better the experimental observations on the LiHoxY1−xF4 compound, with the theoretical results coinciding qualitatively with measurements of the nonlinear susceptibility χ3 In this later case, by increasing gradually, χ3 becomes progressively rounded, presenting a maximum at a temperature T ∗ (T ∗ > Tf ), with both the amplitude of the maximum and the value of T ∗ decreasing gradually. Moreover, we also show that the random field is the main responsible for the smearing of the nonlinear susceptibility, acting significantly inside the paramagnetic phase, leading to two regimes delimited by the temperature T ∗, one for Tf T ∗. It is argued that the conventional paramagnetic state corresponds to T >T ∗, whereas the temperature region Tf < T < T ∗ may be characterized by a rather unusual dynamics, possibly including Griffiths singularities