The van Hemmen model with a transverse field is studied to describe the quantum Ising spin glass. The free energy and phase diagrams (T versus ⍀, and T versus J o /J, where ⍀ is the transverse field, J o and J are the ferromagnetic and random exchange interactions, respectively͒ are calculated for the model with two-peaked and Gaussian exchange distribution. The system presents three ordered phases, namely, spin glass, mixed, and ferromagnetic phases, besides the paramagnetic disordered phase. The influence of the transverse field ⍀ is to destroy the ordered phases. In the (T,⍀) plane our results are compared with those obtained by the replicasymmetry-breaking solution and the same qualitative behavior is observed for the spin-glass transition.The role of quantum fluctuations in spin glass ͑SG͒ remains a long standing theoretical problem. 1 Recently, there has been growing interest in theoretical and experimental investigations of the Ising spin glass in a transverse field to treat the phase transition in quantum spin glasses. 2-6 Experimentally, results for the nonlinear susceptibility provide strong evidence for a finite transition temperature, T c , and as an example we have the so-called proton glasses, 7,8 being a random mixture of ferroelectric and antiferroeletric materials such as Rb 1Ϫx (NH 4 ) x H 2 PO 4 , where proton tunneling in the glass state can be represented by transverse field in the pseudospin Ising model. 9 The transverse Ising spin-glass model has also been used to treat the quantum spin-glass phase transition of the diluted dipole coupled magnet LiHo x Y 1Ϫx F 4 . [10][11][12] The spin-glass problem represents a quite difficult task in statistical mechanics. For many years there has been great controversy on whether the spin-glass transition is of thermodynamic or of dynamic nature. However, simulations 13 and phenomenological scaling arguments at zero temperature 14 suggested the existence of a true thermodynamic phase. Until now, only mean-field models are exactly tractable, but they require sophisticated mathematical tools. 15 The treatment of the quantum spin-glass problem is more complicated than its classical counterpart mainly by two factors: ͑i͒ the system has a dynamic nature from the outset and cannot be simplified to calculation of static quantities even while evaluating statistical mechanical averages; ͑ii͒ quenched disorder and associated very complicated energy landscape resulting in a huge number of local minima of free energy as in the case of the classical spin glass.In the present paper, we study the quantum influence of the transverse field on the spin glass mean-field model introduced by van Hemmen ͑VH͒. 16 This model is exactly soluble, and, unlike the Sherrington-Kirkpatrick 15 model ͑SK͒, its solution does not require the use of the replica trick. It spite of being nonrealistic, mean-field models give a first qualitative understanding of the thermodynamic behavior. Among these are the susceptibility cusp at the freezing temperature T f and the field-induced transition...

The van Hemmen model with a random field is studied to analyze the tricritical behavior in the Ising spin glass phase. The free energy and phase diagram (T versus H and T versus J o /J, where H is the root mean square deviation of the magnetic field, J o and J are the ferromagnetic and root mean square deviation exchange, respectively) are calculated for the model with discrete (or bimodal) and Gaussian distributions. For the case of the bimodal probability distribution (random field and exchange), we have the presence of three ordered phases, namely: spin glass (SG), mixed (∏) and ferromagnetic (F). The root mean square deviation of the random field H destroys the spin glass and mixed phases. The mixed phase doesn't appear with Gaussian distribution. In the plane T versus H, we analyze the tricritical behavior for the case of the bimodal distribution, and we compare it with the results obtained by using the Gaussian distribution that presents only second-order phase transition.

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