Universal Berezinskii-Kosterlitz-Thouless dynamic scaling in the intermediate time range in frustrated Heisenberg antiferromagnets on a triangular lattice

We investigate nonequilibrium properties of the frustrated Heisenberg antiferromagnets on the triangular lattice. Nonequilibrium critical relaxation of frustrated Heisenberg antiferromagnets shows a dynamic transition (or, at least, sharp crossover) near the temperature $T_{\mathrm{u}}=0.282J$ of the equilibrium ${\mathbb{Z}}_2$-vortex unbinding. We show that starting from the high-temperature initial state, due to the presence of ${\mathbb{Z}}_2$ vortices in the system being considered, in a broad temperature range $T<T_{\mathrm{u}}$ the dynamic properties in the intermediate time range are similar to those of the two-dimensional XY model below the Berezinskii-Kosterlitz-Thouless transition. The interaction of ${\mathbb{Z}}_2$ vortices with spin-wave degrees of freedom does not emerge until rather long times.

Figure 1

Projection of spin vectors onto the $xy$ plane for the initial low- and high-temperature states of Monte Carlo simulation.

Figure 2

(a) The time dependence of the correlation length of the system at nonequilibrium evolution from the low-temperature initial state for the linear size $L=240$. (b) The temperature dependence of the equilibrium order parameter $S\left(T\right)$ for various sizes of the system $L$.

Figure 3

(Upper plot) The short-range order parameter starting from the low-temperature (LT) and high-temperature (HT) initial state at $L$ = 240. (Lower plot) Temperature dependence of the equilibrium short-range order parameter for various $L$. The inset shows the behavior in the vicinity of the ${\mathbb{Z}}_2$-vortex unbinding transition (or crossover).

Figure 4

Log-log plot of the function $\xi \left(t\right)$ satisfying Eq. (20); $t$ is measured in units $a^2/\mathrm{\Gamma }$.

Figure 5

Snapshot of relaxation dynamics of system at $T=0.1$ and the times 100 (upper plots), 5000 (middle plots), and 15 000 (lower plots) $\mathrm{MCS}/\mathrm{s}$ from the high-temperature initial state. Left (right) figures—projections of the spins (chirality) vectors onto the $xy$ plane. The radius of each circle on the right figures represents the length of the corresponding chirality vector. Red triangles on the right lower figure denote cores of ${\mathbb{Z}}_2$ vortices; the vortices on the upper and middle figure are not shown (because of their large number, it is impossible to identify them uniquely).

Figure 6

The time dependence of the correlation length of the system at nonequilibrium evolution from the high-temperature initial state for the linear size $L=240$ with ${(t/lnt)}^{1/2}$ (dashed lines) and $t^{1/2}$ (dot-dashed lines) fits.

Figure 7

The time dependence of correlation length $\xi \left(t\right)$ for different system sizes and high-temperature initial state, rescaled according to the dynamic exponent $z=2$ for triangular lattice (upper plot) and square lattice (lower plot). Dashed and dash-dotted lines show the slope of ${(t/lnt)}^{1/2}$ fit.

Figure 8

Time dependence of the autocorrelation spin function for the high-temperature initial state $L=240$, and $T=0.25$ (left plot) and $T=0.35$ (right plot).

Figure 9

(a)–(c) Scaling for the autocorrelation function of the system with $L=240$; $T=0.28$, 0.3, and 0.31 from the high-temperature initial state. (d) The temperature dependence of the anomalous dimension $\eta \left(T\right)$.