We investigate the inverse freezing in the fermionic Ising spin-glass (FISG) model in a transverse field Γ. The grand canonical potential is calculated in the static approximation, replica symmetry and one-step replica symmetry breaking Parisi scheme. It is argued that the average occupation per site n is strongly affected by Γ. As consequence, the boundary phase is modified and, therefore, the reentrance associated with the inverse freezing is modified too. PACS numbers:The inverse transitions (melting or freezing), first proposed by Tammann 1 , are a class of a quite interesting phase transitions, utterly counterintuitive, in which the ordered phase has more entropy than the disordered one 2 . Despite this apparent unconventional thermodynamics, there is now a list plenty of physical systems in which this sort of transition appears (see Ref. 2 and references therein) including, interestingly, high temperature superconductors. 3 In that sense, the search for theoretical models which contain the necessary ingredients to produce such transitions has become a challenging issue as can be seen in Refs. 2 and 4,5,6. However, much less considerations have been given to models in which quantum effects can also be taken into account.

The Sherrington-Kirkpatrick model, in the presence of a Gaussian random field, is studied through the replica formalism within a one-step replicasymmetry-breaking procedure. This model treated in the replica-symmetry approximation does not exhibit a spin-glass phase, since the corresponding order parameter becomes trivially induced by the random field. However, such a phase appears naturally through the present approach, being associated with the onset of replica-symmetry breaking. It is shown that the low-temperature negative-entropy problem, characteristic of the replica-symmetry approximation, is practically resolved within this improved approach. Phase diagrams, the corresponding order parameters, and thermodynamic properties are computed; the present results are expected to be very close to the correct ones, i.e., those that could be obtained from the full replica-symmetry-breaking procedure.

In this work we studied the Hopfield fermionic spin-glass model which allows interpolating from trivial randomness to a highly frustrated regime. Therefore, it is possible to investigate whether or not frustration is an essential ingredient which would allow this magnetic-disordered model to present naturally inverse freezing by comparing the two limits, trivial randomness and highly frustrated regime, and how different levels of frustration could affect such unconventional phase transition. The problem is expressed in the path-integral formalism where the spin operators are represented by bilinear combinations of Grassmann variables. The grand canonical potential is obtained within the static approximation and one-step replica symmetry-breaking scheme. As a result, phase diagrams temperature versus the chemical potential are obtained for several levels of frustration. Particularly, when the level of frustration is diminished, the reentrance related to the inverse freezing is gradually suppressed.

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The infinite-range-interaction Ising spin glass in the presence of a bimodal random field is investigated through the replica method. A one-step replica-symmetry-breaking procedure is considered, and its results are compared to previous ones, obtained in the replica-symmetry approximation. An analysis of the thermodynamic properties at low temperatures shows that the one-step replica-symmetry-breaking produces a substantial amelioration with respect to the replica-symmetric solution; in particular, the low-temperature negativeentropy problem is practically resolved within this improved approach. A curious aspect in replica-symmetric phase diagrams concerns the border of the ferromagnetic phase, which for sufficiently high values of the field may present one or even two tricritical points. In this latter case, it is shown that the higher-temperature tricritical point remains; however, the low-temperature one disappears within the one-step replica-symmetry-breaking procedure, yielding a first-order critical frontier down to zero temperature.