Quantum critical point in the spin glass-antiferromagnetism competition   for fermionic Ising Models
  The competition between spin glass ($SG$) and antiferromagnetic order ($AF$) is analyzed in two sublattice fermionic Ising models in the presence of a transverse $\Gamma$ and a parallel $H$ magnetic fields. The exchange interaction follows a Gaussian probability distribution with mean $-4J_0/N$ and standard deviation $J\sqrt{32/N}$, but only spins in different sublattices can interact. The problem is formulated in a path integral formalism, where the spin operators have been expressed as bilinear combinations of Grassmann fields. The results of two fermionic models are compared. In the first one, the diagonal $S^z$ operator has four states, where two eigenvalues vanish (4S model), which are suppressed by a restriction in the two states 2S model. The replica symmetry ansatz and the static approximation have been used to obtain the free energy. The results are showing in phase diagrams $T/J$ ($T$ is the temperature) {\it versus} $J_{0}/J$, $\Gamma/J$, and $H/J$. When $\Gamma$ is increased, $T_{f}$ (transition temperature to a nonergodic phase) reduces and the Neel temperature decreases towards a quantum critical point. The field $H$ always destroys $AF$; however, within a certain range, it favors the frustration. Therefore, the presence of both fields, $\Gamma$ and $H$, produces effects that are in competition. The critical temperatures are lower for the 4S model and it is less sensitive to the magnetic couplings than the 2S model. 
F. M. Zimmer and S. G. Magalhaes