Elusive Spin-Glass Phase in the Random Field Ising Model

We show rigorously that the spin-glass susceptibility in the random field Ising model is always bounded by the ferromagnetic susceptibility, and therefore that no spin-glass phase can be present at equilibrium out of the ferromagnetic critical line. When the magnetization is, however, fixed to values smaller than the equilibrium value, a spin-glass phase can exist, as we show explicitly on the Bethe lattice.

Figure 1

Free energies $f(m)$ and the stability parameter $\lambda (m)$ versus the magnetization $m$ on a Bethe lattice of degree $c=6$ with Gaussian random fields of unit variance and zero mean. Bottom: We show $T=5$, $T=4.5$, and $T=4.0$ (with $T_c=4.66$); notice the appearance of two minima for $T<T_c$. Middle: A low-temperature case ($T=1$). A spin-glass region appears when the magnetization is fixed to low enough values. Top: The corresponding stability parameter $\lambda (m)$ showing the spin-glass order for $|m|<0.56$.

Figure 2

Phase diagram of the RFIM on the $c=6$ Bethe lattice with Gaussian (left) and bimodal (right) random field. Top panels: Boundary between paramagnetic ($P$) and ferromagnetic ($F$) phases [solid (red) line]. In the low-$T$ region the transition is first order for bimodal fields (red dashed lines are the spinodals). Below the purple dotted line a spin-glass (SG) phase exists for $m=0$. Lower panels: Equilibrium magnetization $m_F$ [darker (blue) line]. A SG phase exists only below the lighter (purple) line.