Replica symmetry breaking in the transverse-field Ising spin-glass model: Two fermionic representations

Abstract

We analyze the infinite range Ising spin glass in a transverse-field $\ensuremath{\Gamma}$ below the critical temperature by a one step replica symmetry breaking theory. The set of $n$ replicas is divided in $r$ blocks of $m$ replicas each. We present results for different values of the block-size parameter $m$. The spin operators are represented by bilinear combinations of fermionic fields and we compare the results of two models: In the four-states (4-S) model the diagonal ${S}_{i}^{z}$ operator has two unphysical vanishing eigenvalues, that are suppressed by a restraint in the two-states (2-S) model. In the static approximation we obtain qualitatively similar results for both models. They both exhibit a critical temperature ${T}_{c}(\ensuremath{\Gamma})$ that decreases when $\ensuremath{\Gamma}$ increases, until it reaches a quantum critical point at the same value of ${\ensuremath{\Gamma}}_{c}$ and they are both unstable under replica symmetry breaking in the whole spin glass phase. Below the critical temperature we present results for the order parameters and free energy.