Nonequilibrium relaxation analysis of a quasi-one-dimensional frustrated $XY$ model for charge-density waves in ring-shaped crystals

We propose a model for charge-density waves in ring-shaped crystals, which depicts frustration between intra- and interchain couplings coming from cylindrical bending. It is then mapped to a three-dimensional uniformly frustrated $XY$ model with one-dimensional anisotropy in connectivity. The nonequilibrium relaxation dynamics is investigated by Monte Carlo simulations to find a phase transition which is quite different from that of usual whisker crystal. We also find that the low-temperature state is a three-dimensional phase vortex lattice with a two-dimensional phase coherence in a cylindrical shell and the system shows power-law relaxation in the ordered phase.

Figure 1

Schematic diagram of ring crystal and mapping onto cuboid.

Figure 2

The local exponent of the order parameter $r_v$ for various temperatures, as a function of $1∕t$. The temperature of each curve is shown in the figure in the unit of $J_a∕k_B(T_c=6.87J_a∕k_B)$. The derivative is calculated after local fitting to a fourth-order polynomial. The anisotropy parameter $\gamma$ equals 32. The horizontal line shows ${\lambda }_c$ obtained by dynamical scaling.

Figure 3

Scaling plot for $\gamma =32$. The temperatures for used data are 6.9, 7.0, 7.1, 7.2, 7.5, 8.0, 8.5, 9.0, 10.0, 11.0, 12.0, and $14.0J_a∕k_B$. The inset shows critical behavior of the relaxation time.

Figure 4

The local exponent for a relatively small anisotropic parameter $\gamma =10$.

Figure 5

Time evolution of phase structure factor $r_{\theta }^b$ (top) and time derivative of $r_{\theta }^b$ multiplied by $t$ (bottom). The derivative is obtained by local fitting to a fourth-order cubic polynomial. The line of slope $-0.22$ means the relaxation at the critical temperature.