Second-order dynamic transition in a $p=2$ spin-glass model

We consider the dynamics of a disordered $p$-spin model with $p=2$, analyzing the dynamics within Ruelle’s thermodynamic formalism, We use an indicator of the dynamical activity to construct the relevant dynamical Gibbs ensemble. We prove that the dynamics in the low-temperature (spin-glass) phase of the model take place at a second-order phase transition between dynamically active and inactive trajectories. We also show that the same behavior is found in a related model of a three-dimensional ferromagnet.

Figure 1

“Phase diagram” associated with the dynamic free energy $\psi \left(s\right)$ of the spin-glass model. The critical point of the model is at $s=0$ and $T=T_c$. The heavy solid line is a second-order phase boundary between ordered and disordered phases. The dashed line is a crossover within the disordered phase. In the high-temperature (normal) regime then the response to a staggered field $h_s$ is positive; the low-temperature (anomalous) regime is characterized by a negative response to this field.

Figure 2

(a) Proposed phase diagram for $h_s>0$. The solid line is a first-order phase transition at which the staggered magnetization and the activity $X$ are discontinuous. It ends at a critical point at $T=T_c$ and $s=0$, but we note that $h_s$ is finite at this critical point. The dashed line is a crossover at which the linear response to a staggered field $h_s$ vanishes. The dashed line is independent of $h_s$ while the solid line approaches the $s=0$ axis as $h_s\to 0$. (b) Behavior as a function of the field $h_s$ in the high-temperature regime. For negative $s$, there is a crossover from normal to anomalous response: at the crossover $X=2$ and $Y$ satisfies Eq. (57). For positive $s$, there is a critical point at $s_c$, with spontaneous symmetry breaking for $s>s_c$. Close to $T_c$, the critical value of $s_c$ is given by Eq. (45). (c) Behavior as a function of $h_s$ in the low-temperature regime. The solid line for $s<0$ is a first-order phase transition between states with positive and negative response to the staggered field, while the line for $s>0$ is the usual first-order transition between spontaneously ordered states.