Tricritical Points in the Sherrington-Kirkpatrick Model in the Presence of Discrete Random Fields

The infinite-range-interaction Ising spin glass is considered in the presence of an external random magnetic field following a trimodal (three-peak) distribution. Such a distribution corresponds to a bimodal added to a probability p0 for a field dilution, in such a way that at each site the field hi obeys P (hi) = p+δ(hi−h0)+p0δ(hi)+p−δ(hi+h0). The model is studied through the replica method and phase diagrams are obtained within the replicasymmetry approximation. It is shown that the border of the ferromagnetic phase may present, for conveniently chosen values of p0 and h0, first-order phase transitions, as well as tricritical points at finite temperatures. Analogous to what happens for the Ising ferromagnet under a trimodal random field, it is verified that the first-order phase transitions are directly related to the dilution in the fields: the extensions of these transitions are reduced for increasing values of p0. Whenever the delta function at the origin becomes comparable to those at hi = ±h0, first-order phase transitions disappear; in fact, the threshold value p∗0, above which all phase transitions are continuous, is calculated analytically as p∗0 = 2(e 3/2 + 2)−1 ≈ 0.30856. The ferromagnetic boundary at zero temperature also exhibits an interesting behavior: for 0 < p0 < p ∗ 0, a single tricritical point occurs, whereas if p0 > p ∗ 0 the critical frontier is completely continuous; however, for p0 = p ∗ 0, a fourthorder critical point appears. The stability analysis of the replica-symmetric solution is performed and the regions of validity of such a solution are identified; in particular, the Almeida-Thouless line in the plane field versus temperature is shown to depend on the weight p0.