Memory effect and phase transition in a hierarchical trap model for spin glasses

We introduce an efficient dynamical tree method that enables us to explicitly demonstrate the thermoremanent magnetization memory effect in a hierarchical energy landscape. Our simulation nicely reproduces the nontrivial waiting-time and waiting-temperature dependences in this nonequilibrium phenomenon. We further investigate the condensation effect, in which a small set of microstates dominates the thermodynamic behavior in the multilayer trap model. Importantly, a structural phase transition of the multilayer tree model is shown to coincide with the onset of the condensation phenomenon. Our results underscore the importance of hierarchical structure and demonstrate the intimate relation between the glassy behavior and structure of barrier trees.

Figure 1

Schematic diagram of hierarchical barrier trees with (a) $\lambda =0.5$ and (b) $\lambda =2.0$. Each node, represented by a circle, of the tree corresponds to either a saddle point or a local minimum. Red closed (open) circles denote the boundary (internal) saddle points, while black circles represent local minima. The backbone of the tree is highlighted by red lines, which also show the connectivity of the saddle points. Note that the level index $l$ increases from top to bottom.

Figure 2

(a) Protocol for temperature variation for simulating memory effect in multi-layer trap model. The system is initially cooled with a constant rate until the temperature reaches $T_w$. The system then stays at $T_w$ for a finite period of waiting time $t_w$ before further cooling to a base temperature. During the subsequent measurement, a small magnetic field is applied to induce magnetization while the system is heated with a constant rate. (b) shows the system energy averaged over many independent runs as a function of time. Panels (c)–(e) show the temperature dependence of zero-field-cooled DC magnetic susceptibility $\chi$, generated by Monte Carlo simulations. The data are taken after waiting during the cooling process with the waiting time $t_w$ in units of $5\times {10}^6$ Monte Carlo steps at (a) $T_w=0.2T_f$, (b) $T_w=0.4T_f$, and (c) $T_w=0.6T_f$.

Figure 3

(a) Memory effect in terms of maximum depth of the relative change of magnetization $(M-M_{\mathrm{ref}})/M_{\mathrm{ref}}$ versus the structural parameter $\lambda =p_{+}/p_{-}$. The red line is a guide to the eye. (b) A few examples of the temperature dependence of $(M-M_{\mathrm{ref}})/M_{\mathrm{ref}}$.

Figure 4

(a) Schematic diagram of a semi-infinite 1D random walk. In this 1D model, all the saddle nodes at the same level are treated identically, which are represented by red dots. (b) Averaged position $\langle x\rangle$ of the walker as a function of time for different value of $\lambda =p_{+}/p_{-}$. Here the average is computed from ${10}^5$ independent Monte Carlo runs.

Figure 5

(a) Participation ratio $Y$ vs temperature for barrier trees obtained from the dynamical tree method. (b) Glassy order parameter $\mathcal{A}$, defined as the area under the $Y\left(T\right)$ curve, vs the tree-structure parameter $\lambda$.