Emergence of U(1) Symmetry in the 3D $XY$ Model with $Z_q$ Anisotropy

We study the three-dimensional $XY$ model with a $Z_q$ anisotropic term. At temperatures $T<T_c$ this dangerously irrelevant perturbation is relevant only above a length scale $\Lambda$, which diverges as a power of the correlation length, $\Lambda \sim {\xi }^{a_q}$. Below $\Lambda$ the order parameter is U(1) symmetric. We derive the full scaling function controlling the emergence of U(1) symmetry and use Monte Carlo results to extract the exponent $a_q$ for $q=4,\dots ,8$. We find that $a_q\approx a_4(q/4{)}^2$, with $a_4$ only marginally larger than 1. We discuss these results in the context of U(1) symmetry at deconfined quantum-critical points separating antiferromagnetic and valence-bond-solid states in quantum spin systems.

Figure 1

$P(m_x,m_y)$ at $h/J=1$ for $q=4$, 8, $L=4$, 32. The temperature $T/J=2.17$ for $Z_4$ and 1.15 for $Z_8$, both less than $T_c/J\approx 2.20$. The size of the histograms corresponds to $m_{x,y}\in [-1,1]$. Angular distributions $P(\theta )$ with $\theta \in [0,2\pi ]$ are shown above each histogram.

Figure 2

Spins in one layer of the $Z_4$ model with $L=10$ at $h/J=1$, $T/J=1.9<T_c$. Here $m_x\approx m_y$, corresponding to $\theta \approx \pi /4$ in $P(r,\theta )$. Arrows are color-coded according to the closest $Z_4$ angle; $n\pi /2$, $n=0$, 1, 2, 3.

Figure 3

The $XY$ order parameter $⟨m⟩$ (solid curves) and the $Z_q$ order parameter $⟨m_q⟩$ (dashed curves) vs temperature for $q=4$, 5, 6. The system sizes are $L=8$, 10, 12, 14, 16, 24, and 32. The curves become sharper (increasing slope) around $T_c$ (indicated by vertical lines). The ratios $h/J$ used are indicated on the graphs.

Figure 4

Scaling of the $Z_q$ magnetization. We use $\sigma =0.52$ for all $q$, and ${\nu }_q$ as indicated in the plots. The colors of the curves correspond to $L$ as in Fig. 3. The dashed curves are the predicted scaling functions with $\tilde{h}$ and prefactors adjusted to fit the data approximately.