Reentrance of Berezinskii-Kosterlitz-Thouless-like transitions in a three-state Potts antiferromagnetic thin film

Using Monte Carlo simulations and finite-size scaling, we study a three-state Potts antiferromagnet on a layered square lattice with two and four layers: $L_z=2$ and 4, respectively. As temperature decreases, the system develops quasi-long-range order via a Berezinskii-Kosterlitz-Thouless transition at finite temperature $T_{c1}$. For $L_z=4$, as temperature is further lowered, a long-range order breaking the $Z_6$ symmetry develops at a second transition at $T_{c2}<T_{c1}$. The transition at $T_{c2}$ is also Berezinskii-Kosterlitz-Thouless-like, but has a magnetic critical exponent $\eta =1/9$ instead of the conventional value $\eta =1/4$. The emergent $U\left(1\right)$ symmetry is clearly demonstrated in the quasi-long-range ordered region $T_{c2}\le T\le T_{c1}$.

Figure 1

Left, staggered magnetization $m_s$ versus $T$ for the three-state AFP model on the two-layer square lattice; the dashed vertical line is set at the critical point $T_{c1}=0.535$. Right, log-log plot of $m_{\mathrm{s}}$ versus $L$ for given temperatures.

Figure 2

Left, plot of $m_s{L}^{1/8}$ versus $T$ for various $L$. Right, an enlarged version of the left plot nearing the critical point. The dashed vertical line is set at the critical point $T_{c1}=0.535$.

Figure 3

Critical exponents $y_{\mathrm{s}},y_{\mathrm{u}}$, and $X_0$ versus $T$ for the three-state AFP model on the two-layer square lattice.

Figure 4

Plot of $\xi /L$ and $C_{\mathrm{v}}$ versus $T$ for various system sizes for the three-state AFP model on the two-layer square lattice; the dashed vertical line is set at the BKT point $T_{c1}=0.535$.

Figure 5

$\xi /L$ versus $T$ for the three-state AFP model on the four-layer square lattice. The dashed vertical line is set at $T_{c1}=0.97$; the solid vertical line is set at $T_{c2}=0.39$.

Figure 6

Left, $m_s$ versus $T$ for the three-state AFP model on the four-layer square lattice; the dashed vertical line is set at the BKT point $T_{c1}=0.97$, and the solid vertical line is set at the BKT point $T_{c2}=0.39$. Right, log-log plot of $m_s$ versus $L$ for the three-state AFP model on the four-layer square lattice; the two dashed lines correspond to the BKT points.

Figure 7

Critical exponents $y_{\mathrm{s}},y_{\mathrm{u}}$, and $X_0$ versus $T$ for the three-state AFP model on the four-layer square lattice.

Figure 8

Plot of $m_s{L}^{y_{\mathrm{s}}-d}$ versus $T$ for the three-state AFP model on the four-layer square lattice. Left, $y_{\mathrm{s}}=15/8$; right, $y_{\mathrm{s}}=35/18$.

Figure 9

Plot of ${\varphi }_6$ and the Binder ratio $Q_{\varphi }$ for the three-state AFP model on the four-layer square lattice. The vertical lines are set at the critical point $T_{c2}=0.39$.

Figure 10

Histogram of $\left({\mathcal{M}}_x,{\mathcal{M}}_y\right)$ for the three-state AFP model on the four-layer square lattice, with system size $L=512$. Left, $T=0.2$; right, $T=0.7$.