Overlap and activity glass transitions in plaquette spin models with hierarchical dynamics

We consider thermodynamic and dynamic phase transitions in plaquette spin models of glasses. The thermodynamic transitions involve coupled (annealed) replicas of the model. We map these coupled-replica systems to a single replica in a magnetic field, which allows us to analyze the resulting phase transitions in detail. For the triangular plaquette model (TPM), we find for the coupled-replica system a phase transition between high- and low-overlap phases, occurring at a coupling ${\varepsilon }^{*}\left(T\right)$, which vanishes in the low-temperature limit. Using computational path sampling techniques, we show that a single TPM also displays “space-time” transitions between active and inactive dynamical phases. These first-order dynamical transitions occur at a critical counting field $s_c\left(T\right)\gtrsim 0$ that appears to vanish at zero temperature in a manner reminiscent of the thermodynamic overlap transition. In order to extend the ideas to three dimensions, we introduce the square pyramid model, which also displays both overlap and activity transitions. We discuss a possible common origin of these various phase transitions, based on long-lived (metastable) glassy states.

Figure 1

(a) Illustration of spins and plaquettes in the TPM. The spins ${\sigma }_i$ are located on the vertices of the lattice. The plaquette variables ${\tau }_{\mu }$ are located on the upward-pointing triangles (shaded). Each plaquette is associated with three spins and the variable ${\tau }_{\mu }$ is given by the product of these spins. (b), (c) Geometrical illustration of the duality relation (14) in the TPM. Panel (b) shows two TPM systems, $a$ and $b$, with coupling as in (3). Panel (c) shows the location of the sites of the dual problem, again two coupled TPMs $a^{*}$ and $b^{*}$. The plaquettes in the dual system bisect the coupling interactions in the direct system, and vice versa.

Figure 2

Illustration of the relation between critical behavior of the Ising model and the TPM. (a) Ising phase diagram. The $h_I=0$ axis is a symmetry line; there is a critical point indicated by a circle, with a first-order (phase coexistence) line for large $J_I$, indicated by a solid line. Selected lines of constant $J_I$ are indicated by dotted lines. (b) The corresponding situation for the TPM in a field. On the solid-and-dashed line (16), the system has a discrete ($Z_2$) symmetry: A critical point and phase coexistence both occur on this line, as indicated. The dotted lines are obtained from (22) for three different values of $h_0$ and correspond to the lines of constant $J_I$ in panel (a). Near the critical point, they indicate the direction of the most relevant renormalization group flow.

Figure 3

Simulations of the TPM in a field. (a) Distribution of the magnetization at various values of $J$ for state points on the self-dual line (17), at system size $L=128$. The bimodal distribution $P\left(m\right)$ indicates a first-order transition, which disappears on reducing $J$. From (7), the same distributions would be obtained when considering the overlap between two coupled TPMs, at appropriate state points. (b) Representative configuration at phase coexistence ($\beta J=2.9$ and $L=128$) showing interfaces between regions of small and large magnetization (corresponding to regions of small and large overlap in the two-replica problem). (c) At our estimated critical point, $(J_c=2.634,h_c=0.072)$ and for various system sizes, we show distributions of the variable $x$ that is obtained by rescaling the order parameter $\mathcal{M}$ to zero mean and unit variance. The solid line is the corresponding result for the 2D Ising model at criticality [26], indicating that the critical point of the TPM in a field (and therefore of the two coupled TPMs) is in the 2D Ising universality class.

Figure 4

Phase diagrams of coupled plaquette models, the 2D TPM (left) and the 3D SPyM (right, see also Sec. 5 below). The solid line corresponds to a line of first-order transitions between a thermodynamic phase of small overlap and one of large overlap between the replicas. This curve is on the self-dual line (17) (dashed line). The first-order transition line ends at a critical point that is in the 2D Ising universality class for the TPM and the 3D Ising universality class for the SPyM. (For the 3D model in a field, the location of the critical point and the form of the phase diagram were already calculated in [16], so we do not repeat this calculation here. The results shown here for the coupled replicas then follow from the mapping in Sec. 2c.)

Figure 5

Average activity $k\left(s\right)$ and the associated susceptibility $\chi \left(s\right)$ in the TPM at $T=0.5$. The main panels show data for system size $L=8$. These results were obtained by the $x$-ensemble method (see text), using trajectories with fixed numbers of events $K$, as shown. On increasing the trajectory length, the crossover from active to inactive behavior becomes increasingly sharp and the susceptibility peak increases. The behavior for smaller systems ($L=4$) is shown in the insets, with both quantities normalized by the system size. In the absence of a phase transition, one expects both $k\left(s\right)/L^d$ and $\chi \left(s\right)/L^d$ to be independent of $L$, so the sharper crossover at $L=8$ is again consistent with an underlying phase transition.

Figure 6

The SPyM consists of spins (gray circles) on the sites of a bcc lattice which interact in quintuplets at the vertices of upward-pointing square pyramids. One such pyramid is indicated; The central spin also participates in four other upward-pointing pyramids whose apexes are the four spins on the upper face of the cube.

Figure 7

Relaxation of the energy of the SPyM at low temperature starting from a random configuration (the system size is $L=16$). The curve shows the characteristic plateaus indicative of hierarchical relaxation, as in the East model and the TPM. (Inset) Average relaxation time as a function of inverse temperature, showing super-Arrhenius behavior.

Figure 8

(a) Activity as a function of $s$ in the SPyM for system size $L=4$, various trajectory lengths, and $T/J=0.65$. The inset shows the corresponding susceptibility. (b), (c) $s$-ensemble phase diagrams for the TPM and SPyM. The solid curves are an estimate of the transition point from the simulations. The dashed lines are extrapolations in the low-temperature regime inaccessible to numerics.

Figure 9

Three-dimensional stack of coupled two-dimensional TPMs.