Avoided criticality and slow relaxation in frustrated two-dimensional models

Frustration and the associated phenomenon of “avoided criticality” have been proposed as an explanation for the dramatic relaxation slowdown in glass-forming liquids. To test this, we have undertaken a Monte Carlo study of possibly the simplest such problem, the two-dimensional $XY$ model with frustration corresponding to a small flux $f$ per plaquette. At $f=0$, there is a Berezinskii-Kosterlitz-Thouless transition at $T^{*}$, but at any small but nonzero $f$, this transition is avoided and replaced (presumably) by a vortex-ordering transition at much lower temperatures. We thus have studied the evolution of the dynamics for small and moderate $f$ as the system is cooled from above $T^{*}$ to below. Although we do find strongly temperature-dependent slowing of the dynamics as $T$ crosses $T^{*}$ and that simultaneously the dynamics becomes more complex, neither effect is anywhere nearly as dramatic as the corresponding phenomena in glass-forming liquids. At the very least, this implies that the properties of supercooled liquids must depend on more than frustration and the existence of an avoided transition.

Figure 1

Spin autocorrelation function $C_{ss}\left(t\right)$ versus ${log}_{10}\left(t\right)$ for the unfrustrated ($f=0$) 2D $XY$ model at temperatures from $T/J=1$ to 0.1 (from left to right: $T/J=1.0,0.9,0.8,0.7,0.6,0.5,0.4,0.3,0.2,0.1$) for different system sizes from $L=10$ to $L=40$. System size $L=34$ includes one additional curve at $T/J=0.13$.

Figure 2

Log-log plot of $C_{ss}\left(t\right)$ at short times (first regime of the data in Fig. 1) for the unfrustrated ($f=0$) 2D $XY$ model with $L=40$ at several temperatures: $T/J=0.1,0.2,0.3,0.4,0.5$ (from top to bottom). Inset: Extracted slope versus temperature. The dashed line is the predicted behavior $1/\left(4\pi \right)(T/J)$.

Figure 3

Log-linear plot of $C_{ss}\left(t\right)$ at long times (second regime of the data in Fig. 1) for the unfrustrated ($f=0$) 2D $XY$ model with $L=40$ at several temperatures $T/J=0.2,0.3,0.4,0.5$ from top to bottom, showing the exponential decay at long times.

Figure 4

Arrhenius plot of the $T$ dependence of the relaxation time, i.e., ${log}_{10}{\tau }_2$ versus $J/T$, for the unfrustrated ($f=0$) 2D $XY$ model for $L=10,20,34,40$ (from bottom to top). Inset: Plot of ${log}_{10}{\tau }_2$ versus ${log}_{10}(J/T)$ for $T\lesssim 0.5J$, showing a power-law behavior compatible with $T^{-3/2}$ (displayed as the dashed straight line). $J/T^{*}$ denotes the BKT transition temperature.

Figure 5

Finite-size scaling in the unfrustrated ($f=0$) 2D $XY$ model for $L=10,20,34,40\text{:}{\tau }_2$ versus $L^2$ for a low temperature of $T/J=0.2$.

Figure 6

Spin autocorrelation function $C_{ss}\left(t\right)$ versus ${log}_{10}\left(t\right)$ for the frustrated 2D $XY$ model at several temperatures from $T/J=2$ down to $T/J=0.1$ (from left to right: $T/J=2.0,1.8,1.6,1.4,1.2,1.0,0.9,0.8,0.7,0.6,0.5,0.4,0.3,0.2,0.13,0.1$) for five frustrations: $f=13/34,5/34,1/34,10/{\left(34\right)}^2$, and $5/{\left(34\right)}^2$. Frustration $f=13/34$ includes one additional curve at a temperature of $T/J=0.16$.

Figure 7

Test of the so-called time-temperature superposition principle: Same data as in Fig. 6 plotted versus the logarithm of the scaled time $t/\tau \left(T\right)$. The data are divided into “high $T$” and “low $T$” as described in the text. Although superposition is never found for the three smallest frustrations, it is observed approximately for the two strongest at high temperatures and then violated at low temperatures.

Figure 8

Arrhenius plot of ${log}_{10}\left(\tau \right)$ versus $J/T$ for $f=5/{\left(34\right)}^2,10/{\left(34\right)}^2,1/34,5/34,13/34$ (from top to bottom). $J/T^{*}$ denotes the BKT transition temperature.

Figure 9

Log-log plot of $\tau$ versus $J/T$. Same data as in Fig. 8: $f=5/{\left(34\right)}^2,10/{\left(34\right)}^2,1/34,5/34,13/34$ (from top to bottom). The dashed lines are linear fits to the data with slopes given in the figure. Inset: Zoom on the low-$T$ region for the two largest frustrations $f=5/34$ and $f=13/34.J/T^{*}$ denotes the BKT transition temperature.

Figure 10

Number of vortices $N$ minus the number of irreducible vortices $N_0$ for $f=1/34,5/34,13/34$ and for $f=0$ with $L=40$ (the number of irreducible vertices is 34, 170, 442, and 0, respectively). At $T\approx 0.7J$ and below the thermal vortices have virtually disappeared. $T^{*}/J$ denotes the BKT transition temperature.

Figure 11

Comparison of the time-dependent spin and current autocorrelation functions for $f=1/34$ and $T/J=0.3$. The latter has a faster initial decay, but the relaxation times describing the final relaxation stage are comparable.

Figure 12

Effect of pinning the frustration-induced vortices in the uniformly frustrated $XY$ model for $f=1/34$ at several temperatures, $T/J=0.6,0.5,0.3$: Time-dependent current (top) and spin (bottom) autocorrelation functions in the system with and without vortex pinning. The solid lines are with vortex pinning, and the dashed lines are without vortex pinning.