The random-field Ising model, with a trimodal distribution (sum of three b functions), is studied within the mean-field approximation.The averaged free energy and order parameter are analyzed in detail, numerically and through the Landau expansion, to construct the complete phase diagram in the parameter space of the system. A line of tricritical points, ending at a vestigial tricritical point, is obtained, a result also suggested recently by Mattis. Some new features of the phase diagram are also reported. P(H;) =(2a ) ' exp( -H /cr ) (2) led to different phase diagrams. For the bimodal distribution a tricritical point was observed at H, =0.44 and T] 0.67, and for 0.44~H & 0.5 the transition was found to be of first order. On the other hand, the Gaussian distribution led to a second-order phase transition down to T 0. During recent years some papers have appeared which discuss the criteria for the determination of the order of the low-temperature phase transition and its dependence on the form of the field distribution. Recently, Houghton, Khurana, and Seco have performed detailed numerical studies based on the analysis of the hightemperature series for the static susceptibility of the RFIM with a Gaussian distribution and have observed fluctuation-driven first-order transitions at low temperatures, for d &4. These calculations emphasized the importance of the dimensionality and the form of the I In recent years the random-field Ising model (RFIM) has been a topic of much theoretical and experimental discussion. ' One of the main points of controversy was the lower critical dimension dg for the existence of a ferromagnetic phase transition. In the last three years various theoretical studies have concluded that d~-2. Other important questions about the RFIM have been the existence of a tricritical point, the order of low-temperature phase transitions, and the dependence of the phase diagram on the form of the random-field distribution. In earlier work on the RFIM, it was observed that, within the mean-field approximation, the bimodal distribution P (H; ) = , ' [b(Hi -H ) +-b(H;+H ) i, and the Gaussian distribution random-field distribution in the determination of the nature of phase transitions in the RFIM. In a comment on the work of Houghton et aL, Mattis used the mean-field approach to examine the possibility of the existence of a tricritical point for the RFIM with a trimodal distribution, P (H; ) =pb(H; ) + 2 (1 -p ) (b(H; -H ) +b(H; +H ) ], (3) and observed that the phase diagram exhibited tricritical points for the range of values 0~p~0.25. Considering that the case p = -, ' of Eq.(3) can be taken as a good approximation to a Gaussian distribution, Mattis concluded that there is no tricritical point for a Gaussian distribution. In this short note we report a calculation of the complete phase diagram of the RFIM with a trimodal distribution, Eq.(3). We use the mean-field approximation and the Landau expansion of the averaged free energy to analyze the order of the phase transitions. Detailed and careful nu...
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