Caution on emergent continuous symmetry: A Monte Carlo investigation of the transverse-field frustrated Ising model on the triangular and honeycomb lattices

Continuous symmetries are believed to emerge at many quantum critical points in frustrated magnets. In this work, we study two candidates of this paradigm: the transverse-field frustrated Ising model (TFFIM) on the triangular and the honeycomb lattices. The former is the prototypical example of this paradigm, and the latter has recently been proposed as another realization. Our large-scale Monte Carlo simulation confirms that the quantum phase transition (QPT) in the triangular lattice TFFIM indeed hosts an emergent O(2) symmetry, but that in the honeycomb lattice TFFIM is a first-order QPT and does not have an emergent continuous symmetry. Furthermore, our analysis of the order-parameter histogram reveals that such different behavior originates from the irrelevance and relevance of anisotropic terms near the QPT in the low-energy effective theory of the two models. The comparison between theoretical analysis and numerical simulation in this work paves the way for scrutinizing investigation of emergent continuous symmetry at classical and quantum phase transitions.

Figure 1

(a) Triangular lattice. $a$, $b$, and $c$ are three sublattices. $J$ is the nearest-neighbor antiferromagnetic coupling. ${\mathbf{r}}_1$ and ${\mathbf{r}}_2$ are the primitive vectors of the magnetically ordered clock phase. (b) Honeycomb lattice. $a$ and $b$ are the sublattices and ${\mathbf{r}}_1$ and ${\mathbf{r}}_2$ are the primitive vectors of the honeycomb lattice. $J_1$, $J_2$, and $J_3$ are the nearest-, next-nearest-, and third-nearest-neighbor antiferromagnetic couplings, respectively. The green dashed lines indicate the three degenerate ordered states, which are translational invariant along the lines. For example, the spin order in (b) is translation invariant along the green dashed line in parallel to the ${\mathbf{r}}_1$ direction.

Figure 2

(a) Chart for the order parameter of the triangular lattice TFFIM: $(m_1,m_2,m_3)$ are the three-sublattice magnetization, separated by an angle of $\frac{2\pi }{3}$. (b) Chart for the order parameter of the honeycomb lattice TFFIM: $\vec{m}=(m_1,m_2,m_3)$ is a $\text{O}\left(3\right)$ vector of the three-sublattice magnetization. It is presented in a spherical coordinate.

Figure 3

Upper row: histograms of the order parameter for the triangular lattice TFFIM at $h=0.4$ and 1.5, with $L=6$. Lower row: histogram of the order parameter for the triangular lattice TFFIM with same set of $h$ for $L=12$. According to the crossing of the Binder cumulant in Fig. 4, the quantum critical point is at $h_c=1.64\left(1\right)$. Left panels show clearly sixfold rotational symmetry and the right panels show emergent $\text{O}\left(2\right)$ symmetry close to the QCP.

Figure 4

Binder cumulant $U$ as function of transverse field $h$ for different system sizes for the triangular lattice TFFIM. Inset: data collapse of the $U$ for $h_c=1.64\left(1\right)$ and $\nu =0.67\left(2\right)$.

Figure 5

Upper row: histograms of the order parameter for the honeycomb lattice TFFIM at $h=2.45$ and 2.5 for $L=6$. Lower row: histogram of the order parameter for the honeycomb lattice TFFIM with same set of $h$ for $L=8$. There is $no$ sign of emergent continuous $\text{O}\left(3\right)$ symmetry. Note, according to the crossing of the Binder cumulant in Fig. 6, the quantum phase transition is at $h_c=2.48\left(1\right)$. The histograms at $h=2.5$ clearly contain the coexistence of the discrete symmetry breaking at $h<h_c$ and the zero magnetization at $h>h_c$.

Figure 6

Binder cumulant for the honeycomb lattice TFFIM. As system size increases, it is clear that $U$ develops a negative peak that grows with increasing $L$, indicating a first-order phase transition at $h=2.48\left(1\right)$.

Figure 7

Anisotropic term ${\nu }_6$ as function of transverse field $h$ for different system sizes on the triangular lattice TFFIM. ${\nu }_6$ goes to zero at the QCP, giving rise to the emergent continuous $\text{O}\left(2\right)$ symmetry. The blue vertical dashed line highlights the position of $h_c$. The blue horizontal dashed line highlights the value of ${\nu }_6=0$.

Figure 8

${\nu }_4$ (a) and ${\nu }_6$ (b) as function of transverse field $h$ for different system sizes on the honeycomb lattice TFFIM. As $h$ is approaching the $h_c$, ${\nu }_4$ and ${\nu }_6$ increase greatly, in obvious contrast to the anisotropic term in the triangular lattice case (see Fig. 7), and give rise to a first-order phase transition (see Figs. 5 and 6). The blue vertical dashed line highlights the position of $h_c$.