Scaling dimensions of higher-charge monopoles at deconfined critical points

The classical cubic dimer model has a columnar ordering transition that is continuous and described by a critical Anderson-Higgs theory containing an $\mathrm{SU}\left(2\right)$-symmetric complex field minimally coupled to a noncompact $\mathrm{U}\left(1\right)$ gauge theory. Defects in the dimer constraints correspond to monopoles of the gauge theory, with charge determined by the deviation from unity of the dimer occupancy. By introducing such defects into Monte Carlo simulations of the dimer model at its critical point, we determine the scaling dimensions $y_2=1.48\pm 0.07$ and $y_3=0.20\pm 0.03$ for the operators corresponding to defects of charge $q=2$ and 3, respectively. These results, which constitute the first direct determination of the scaling dimensions, shed light on the deconfined critical point of spin-$\frac{1}{2}$ quantum antiferromagnets, thought to belong to the same universality class. In particular, the positive value of $y_3$ implies that the transition in the $JQ$ model on the honeycomb lattice is of first order.

Figure 1

Illustration of a local charge $q$, consisting of a site with $q+1$ intersecting dimers. Charge $q=2$ and 3 occur in the configurations shown above and their rotations.

Figure 2

Initial step of the update process for background dimers. The link monomer and the monomer at the starting node ${\vec{r}}_0$ are indicated by red and blue dots. The arrows in the figure indicate the direction of the link monomer.

Figure 3

Examples of individual hopping steps of the link monomer (blue dot). The first two rows show the scenario when the node ahead has no charge. The third and fourth lines show the step where the node ahead has a charge 2. Only hopping steps that do not change the local charge are allowed. These steps can be generalized to the case where the node has higher $q$.

Figure 4

An example of a termination step for the background dimer update. When the link monomer returns to the starting point ${\vec{r}}_0$ of the update loop, it can annihilate the monomer on the starting site and result in a new configuration in ${\mathcal{C}}_q({\vec{r}}_1,{\vec{r}}_2)$.

Figure 5

Examples of the update step that allow a charge-2 monopole to be moved. The location of the charge is marked with red dot. Each possible translation of the charge involves moving exactly two of the three dimers constituting the monopole. Depending on the orientation of the dimers at the charge, the number of ways to hop out of the initial state can differ.

Figure 6

The MC procedure can be thought of as forming a graph with edges connecting the configurations that are related by a single ${\mathcal{T}}_2$ MC step. Thus, transition probabilities $p_{a\to b}$ and $p_{b\to a}$ are nonzero iff they are adjacent on this graph. The number of transitions out of (or into) a configuration $a$ gives the degree $n_a$. Transition probabilities need to be picked such that configurations are sampled with the desired Boltzmann probabilities.

Figure 7

Examples of the update step that allow a charge-3 monopole to move. When the dimers at the charge $q=3$ are in a noncoplanar configuration, three out of the four dimers lie on adjacent edges of a unit cube. There are two such triplets and two associated cubes. When there is a dimer at an edge diagonally opposite to any of these three dimers, as shown above, there are two possible transitions of the charge out of the configuration. Note that such transitions always result in a noncoplanar arrangement of dimers.

Figure 8

Monopole distribution function ${\mathcal{G}}_q$ for a pair of charges with $q=\pm 2$. The two figures show the data for different four-dimer interactions $v_4=1.0$ (top), $1.5$ (middle), and 10 (bottom). The functions have been rescaled with $L^{2x_q}$ where the scaling dimension $x_q$ is related to the RG eigenvalue $y_q$ obtained from Fig. 9 by $x_q=d-y_q$. The dashed lines have slope $-2x_q$, and are expected to be parallel to the scaled correlation for $1\ll R\ll L$.

Figure 9

Log-log plot of ${\mathcal{G}}_2(R=\rho R_{\max },R^{\prime }=1,L)$ for various system sizes at the critical temperature, with $v_4=1.0$ (top), $1.5$ (middle), and 10 (bottom), and $\rho =0.95$. The RG eigenvalue $y_2$ is related to the slopes $s$ of the lines as $s=-2x_q=-2(d-y_q)$.

Figure 10

Top: pair correlation function for charge $q=3$, using piecewise estimation. As for the case of charge 2 in Fig. 8, the function has been rescaled to show the scaling with $L$. The dashed line has a slope of $-2x_3$. Bottom: log-log plot of ${\mathcal{G}}_2(R=\rho L,R^{\prime }=6,L)$ versus $L$. The scaling dimension can be inferred directly from the slope of the best-fit line.

Figure 11

Pair correlation function ${\mathcal{G}}_3(R=\rho L,R^{\prime }=1,L)$ calculated with an additional repulsive interaction between the charges of the form given in Eq. (25) with $\theta =\frac{5}{2}$.

Figure 12

Log-log plot of ${\mathcal{G}}_3(R=\rho L,R^{\prime }=1,L)$ versus $L$, for $v_4=1.5$ (left) and $10.0$ (right) and $\rho =0.95$, calculated with an added repulsive potential. The continuous lines show an error-weighted least-square fit. The dotted and dashed lines correspond to the cases where $y_3$ would have been 0 and $0.5$, respectively. The RG eigenvalue $y_3$ is related to the slope $s$ by $s=-2(d-\theta -y_3)$, where $\theta =\frac{5}{2}$ parametrizes the additional interaction.