The Ising spin glass in a transverse field revisited. Results of two fermionic models

We analyze the infinite range Ising spin glass in a transverse field Γ by using Grassmann variables in a field theory where the spin operators are represented by bilinear combinations of fermionic fields. We compare the results of two fermionic models. In the four state (4S)-model the diagonal Siz operator has two vanishing eigenvalues, that are suppressed by a restraint in the two states (2S)-model. Within a replica symmetric theory and in the static approximation we obtain qualitatively similar results for both models. They both exhibit a critical temperature Tc(Γ) that decreases when Γ increases, until it reaches a quantum critical point at the same value of Γc and they are both unstable under replica symmetry breaking in the whole spin glass phase. Our results are in agreement with recent renormalization group calculations in three-dimensional systems.