Heterogeneous freezing in a geometrically frustrated spin model without disorder: Spontaneous generation of two time scales

By considering the constrained motion of classical spins in a geometrically frustrated magnet, we find a dynamical freezing temperature below which the system gets trapped in metastable states with a “frozen” moment and dynamical heterogeneities. The residual collective degrees of freedom are strongly correlated, and by spontaneously forming aggregates, they are unable to reorganize the system. The phase space is then fragmented in a macroscopic number of disconnected sectors (broken ergodicity), resulting in self-induced disorder and “thermodynamic” anomalies, measured by the loss of a finite configurational entropy. We discuss these results in view of experimental results on the kagome compounds, SrCr${}_{9p}$Ga${}_{12-9p}$O${}_{19}$, (H${}_3$O)Fe${}_3$(SO${}_4$)${}_2$(OH)${}_6$, Cu${}_3$V${}_2$O${}_7$(OH)${}_2·2{\text{H}}_2$O, and Cu${}_3$BaV${}_2$O${}_8$(OH)${}_2$.

Figure 1

Simplest motion compatible with the constraint: color exchange along a loop of length $L$ (e.g., $L=6$). Note that the flip of the central loop (on the left, before the move) facilitates the motion of neighboring sites by creating a new flippable loop (on the right, after the move).

Figure 2

Spin autocorrelation as a function of time (Monte Carlo sweeps) with decreasing $T$ (from left to right), $C\left(0\right)=1$ ($L=144$). Inset: long-time tail (rescaled), $1/t^{2/3}$ (solid line), as described by a height model.

Figure 3

Two relaxation time scales.

Figure 4

Frozen moment squared as a function of temperature and observation time, Eq. (10). $L=144$.

Figure 5

Real-space picture of the autocorrelation, $C_i\left(t\right)={\mathbf{S}}_i\left(t\right)·{\mathbf{S}}_i\left(0\right)$, at time $t={10}^3$ and $T=0.1$. Black, frozen sites; white, $C_i\left(t\right)=-1/2$; gray, the sites which have moved between 0 and $t$ but have returned to their initial state [$C_i\left(t\right)=1$].

Figure 6

Distribution of the sizes of the frozen clusters. The average is $\langle s\rangle =42$ sites (inset: finite-size effect) or the emergent length scale is ${\langle s\rangle }^{1/2}$. The dashed line is a guide to the eye (exponential).

Figure 7

The radial distribution function of active degrees of freedom: probability to have a flippable hexagon at distance $r$ from a given flippable hexagon at 0 (normalized by the number of hexagonal sites) averaged over the uniform ensemble. While the nearest-neighbor position is not compatible with the constraint, the first peak corresponds to an attraction of next-nearest-neighbor hexagons.

Figure 8

“Jammed” cluster (the smallest one, in gray): no 6-loop can unjam any of its 12 sites. The shortest “unjamming” loop (of length 10) is shown (dashed line).

Figure 9

Dynamical heterogeneities in space. The gray scale is proportional to the local frequency $f_i$ of the site from black (frozen) to white (high frequency). $t={10}^4$ and $L=144$.

Figure 10

Nonuniform slowing down of the dynamics by lowering the temperature. $T>T_d$, a homogeneous (Gaussian) distribution of local frequencies; $T<T_d$, a heterogeneous (skewed) distribution; and $T<T_g$, a frozen fraction appears. $t={10}^4$ and $L=144$.

Figure 11

Hierarchical structure of the phase space and exponential number of disconnected sectors $N_t$ at low temperatures (solid circles). $N_t$ corresponds also to the number of zero eigenvalues of the matrix $\mathbf{w}$. The total number of states $N_c$ is denoted by squares; topological sectors $N_{\text{topo}}$ (diamond) result from the complete enumeration of states on clusters of size $N=27,36,81,108$.

Figure 12

Spectral function at different $T$ (dashed lines are ${\omega }^{-\alpha }$).

Figure 13

Distribution of local field strengths at the centers of the hexagons. Evolution from high $T$ to $T<T_g$. A typical static state (no loop flip) is given for comparison (peaks at $0,\sqrt{3},3$; see top).