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 ${p}_{0}$ for a field dilution, in such a way that at each site the field ${h}_{i}$ obeys ${P(h}_{i}{)=p}_{+}\ensuremath{\delta}{(h}_{i}\ensuremath{-}{h}_{0}{)+p}_{0}\ensuremath{\delta}{(h}_{i}{)+p}_{\ensuremath{-}}\ensuremath{\delta}{(h}_{i}{+h}_{0}).$ The model is studied through the replica method and phase diagrams are obtained within the replica-symmetry approximation. It is shown that the border of the ferromagnetic phase may present, for conveniently chosen values of ${p}_{0}$ and ${h}_{0},$ 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 ${p}_{0}.$ Whenever the $\ensuremath{\delta}$ function at the origin becomes comparable to those at ${h}_{i}=\ifmmode\pm\else\textpm\fi{}{h}_{0},$ first-order phase transitions disappear; in fact, the threshold value ${p}_{0}^{*},$ above which all phase transitions are continuous, is calculated analytically as ${p}_{0}^{*}{=2(e}^{3/2}{+2)}^{\ensuremath{-}1}\ensuremath{\approx}0.30856.$ The ferromagnetic boundary at zero temperature also exhibits an interesting behavior: for $0l{p}_{0}l{p}_{0}^{*},$ a single tricritical point occurs, whereas if ${p}_{0}g{p}_{0}^{*}$ the critical frontier is completely continuous; however, for ${p}_{0}{=p}_{0}^{*},$ a fourth-order critical point appears. 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 ${p}_{0}.$