Fate of $\mathbb{C}{\mathbb{P}}^{N-1}$ Fixed Points with $q$ Monopoles

We present an extensive quantum Monte Carlo study of the Néel to valence-bond solid (VBS) phase transition on rectangular- and honeycomb-lattice SU($N$) antiferromagnets in sign-problem-free models. We find that in contrast to the honeycomb lattice and previously studied square-lattice systems, on the rectangular lattice for small $N$, a first-order Néel-VBS transition is realized. On increasing $N\ge 4$, we observe that the transition becomes continuous and with the same universal exponents as found on the honeycomb and square lattices (studied here for $N=5$, 7, 10), providing strong support for a deconfined quantum critical point. Combining our new results with previous numerical and analytical studies, we present a general phase diagram of the stability of ${\mathbb{C}\mathbb{P}}^{N-1}$ fixed points with $q$ monopoles.

Figure 1

(a) Deconfined RG flow diagram for SU($N$) antiferromagnets with $q$-fold degenerate VBS phases, in the field theoretic space of monopole fugacity for $q$ monopoles (${\lambda }_q$) and the tuning parameter $g$ of the critical point (see the text and Ref. 10 for details). In this work, we give a complete phase diagram in $q\mathrm{\text{−}}N$ space for which this RG flow diagram can be realized (see Table ). (b)–(d) Couplings of Eq. (1): (b) The honeycomb lattice with $J_1$ and $J_2$. (c) The rectangular lattice with $J_1^x$, $J_1^y$, and $J_2$. (d) The $Q$ interaction shown is used here only on the rectangular systems. The $A$ and $B$ sublattices (black and white sites) have SU($N$) spins transforming in the fundamental and conjugate to fundamental representations, respectively.

Figure 2

First-order transition for $q=2$ and $N=3$ [rectangular lattice with SU(3) spins]. Magnetic susceptibility for SU(3) on the rectangular lattice. The sharp jump is indicative of a first-order transition. The inset shows a double-peaked histogram of data taken from a point in the middle of the transition ($J_1^x/Q^{x,x}=2.71$) for $L=48$, thus providing further evidence for a first-order transition. To accommodate the rectangular-lattice symmetry [41], we take a lattice with $L_x=4L_y/3$; $L=L_y$ in our legend.

Figure 3

Continuous transition for $q=2$ and $N=7$ [rectangular lattice with SU(7) spins]. (a) This panel shows the Binder ratio data. (b) Both the magnetic (blue squares) and VBS (green circles) susceptibility data. The data have been collapsed such that ${\mathbb{Y}}_N(z)=L^{1+{\eta }_N}{\chi }_N(z)+(a+bz)L^{-\omega }$ and ${\mathbb{Y}}_V(z)=L^{1+{\eta }_V}{\chi }_V(z)$ with ${\eta }_N=0.639$, $a=8.5$, $b=0.1$, $\omega =0.5$, and ${\eta }_V=1.26$. Also, $z=[(g-g_c)/g_c]L^{1/\nu }$ with $g=J_2/J_1^x$, $g_c=0.7552$, and $\nu =0.69$. For the magnetic susceptibility, the following system sizes were used in the collapse: $L=42$, 48, 54, 60, 66, 72, 78, 84, 90, 96, 102, 108. For the VBS susceptibility, the following system sizes were used in the collapse: $L=36$, 42, 48, 54, 60, 66.

Figure 4

Continuous transition for $q=3$ and $N=7$ [honeycomb lattice with SU(7) spins]. (a) The Binder ratio. (b) Both the magnetic (blue squares) and VBS (green circles) susceptibility data. The data have been collapsed such that ${\mathbb{Y}}_N(z)=L^{1+{\eta }_N}{\chi }_N(z)+(a+bz)L^{-\omega }$ and ${\mathbb{Y}}_V(z)=L^{1+{\eta }_V}{\chi }_V(z)$ with ${\eta }_N=0.67$, $a=20.0$, $b=0.8$, $\omega =1.0$, and ${\eta }_V=1.41$. Also, $z=[(g-g_c)/g_c]L^{1/\nu }$ with $g=J_2/J_1$, $g_c=0.5196$, and $\nu =0.72$. For the magnetic susceptibility, the following system sizes were used in the collapse: $L=36$, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96. For the VBS susceptibility, the following system sizes were used in the collapse: $L=18$, 24, 30, 36, 42, 48, 54. There are $2L^2$ lattice sites.

Figure 5

Comparison of anomalous dimensions of Néel and VBS operators in the case of continuous transitions for $q=2$, 3, and 4. (a) Anomalous dimension of the Néel order parameter as a function of $1/N$. (b) Anomalous dimension of the VBS order parameter as a function of $1/N$. The gray squares are the results of a previous square-lattice study ($q=4$) [16, 18]. The blue circles are new results from the honeycomb lattice ($q=3$), and the green diamonds are new results from the rectangular lattice ($q=2$). The red line is the $1/N$ expansion. The agreement of the new data with both the $q=4$ data as well as the $1/N$ computation is striking.