Behavior of Ising Spin Glasses in a Magnetic Field

We study the existence of a spin-glass phase in a field using Monte Carlo simulations performed along a nontrivial path in the field-temperature plane that must cross any putative de Almeida–Thouless instability line. The method is first tested on the Ising spin glass on a Bethe lattice where the instability line separating the spin glass from the paramagnetic state is also computed analytically. While the instability line is reproduced by our simulations on the mean-field Bethe lattice, no such instability line can be found numerically for the short-range three-dimensional model.

Figure 1

Magnetic field $H$ versus temperature $T$ phase diagram of the Ising spin glass on a mean-field random regular graph with connectivity $c=6$ with Gaussian interactions computed using the cavity method. The paramagnetic (PM) phase is separated from the spin-glass (SG) phase by the de Almeida–Thouless line [red (or light gray) curve]. The diagonal [blue (or dark gray)] line $h(T)$ represents the simulation path followed in the Monte Carlo simulations. Because the path starts at $T>T_c$ for $H=0$ and then increases along a diagonal $h(T)=r(T-T_{\min })$ in the $H\mathrm{\text{−}}T$ plane above $T_{\min }\ll T_c$ up to a value beyond $T_c$ (dashed vertical line), an intersection with a putative AT line is guaranteed. The same approach is used for the diluted 3D simulations where $T_c=0.663(6)$ [22].

Figure 2

Rescaled susceptibility $\chi /N^{1/3}$ from Monte Carlo data of the regular random graph model with Gaussian disorder and connectivity $c=6$. Left: Data for zero-field cross at the analytically predicted critical temperature $T_c=1.807$. Right: Data along $h_{\mathrm{BL}}(T)=1.0(T-0.5)$. The crossing point agrees very well with the values in Fig. 1.

Figure 3

Finite-size correlation length as a function of $T/T_c$ for the 3D Ising spin glass at zero field with bimodal interactions. The data cleanly cross at $T_c=0.663(6)$.

Figure 4

Finite-size correlation length as a function of $h_{3\mathrm{D}}(T)=0.7(T-0.3)$ (finite-field branch in Fig. 1) for the 3D Ising spin glass. The apparent crossing of the data wanders towards zero for increasing system size (see inset for a zoom into the low-field region); i.e., there is no sign of a transition for $H>0$ (arrows mark crossings between $L$ pairs).