  The stability of spin-glass (SG) phase is analyzed in detail for a fermionic
Ising SG (FISG) model in the presence of a magnetic transverse field $\Gamma$.
The fermionic path integral formalism, replica method and static approach have
been used to obtain the thermodynamic potential within one step replica
symmetry breaking ansatz. The replica symmetry (RS) results show that the SG
phase is always unstable against the replicon. Moreover, the two other
eigenvalues $\lambda_{\pm}$ of the Hessian matrix (related to the diagonal
elements of the replica matrix) can indicate an additional instability to the
SG phase, which enhances when $\Gamma$ is increased. Therefore, this result
suggests that the study of the replicon can not be enough to guarantee the RS
stability in the present quantum FISG model, especially near the quantum
critical point. In particular, the FISG model allows changing the occupation
number of sites, so one can get a first order transition when the chemical
potential exceeds a certain value. In this region, the replicon and the
$\lambda_{\pm}$ indicate instability problems for the SG solution close to all
range of first order boundary.
