Weak Plaquette Valence Bond Order in the $S\mathbf{=}1/2$ Honeycomb $J_1\mathbf{-}{J}_2$ Heisenberg Model

Using the density matrix renormalization group, we investigate the $S=1/2$ Heisenberg model on the honeycomb lattice with first ($J_1$) and second ($J_2$) neighbor interactions. We are able to study long open cylinders with widths up to 12 lattice spacings. For $J_2/J_1$ near 0.3, we find an apparently paramagnetic phase, bordered by an antiferromagnetic phase for $J_2\lesssim 0.26$ and by a valence bond crystal for $J_2\gtrsim 0.36$. The longest correlation length that we find in this intermediate phase is for plaquette valence bond order. This correlation length grows strongly with cylinder circumference, indicating either quantum criticality or weak plaquette valence bond order.

Figure 1

(a) One hexagon of the honeycomb lattice showing $J_1$ and $J_2$ interactions. The six sides of the hexagon are $J_1$ bonds, and the six $J_2$ bonds are shown as dashed (blue) lines. (b) Illustration of the pattern of PVB order. The spin-spin correlations differ between the bonds shown as thin (black) lines versus thick (red) lines. (c) Three phases on the $YC$6-0 cylinder with 300 sites and a gradient of $J_2$. The widths of lines are proportional to $|⟨S_i·S_j⟩|$. We also show the second-neighbor spin correlation but only when its value is negative. The arrows represents the values of $⟨S_i^z⟩$ at each site. The scales for correlations and magnetizations are indicated. We remove several bonds in one zigzag row of the cylinder, so that the AF spin pattern is more clearly seen. The two vertical lines at $J_2=0.26$ and 0.36 indicate estimates of the phase boundaries.

Figure 2

The staggered magnetization at the center of the cylinder versus $J_2$ for various $XC$ cylinders. The inset shows how the local magnetization decays from the edge of a long $XC$10 cylinder for various values of $J_2$.

Figure 3

The entanglement entropy and its derivative versus $J_2$ for different $XC$ cylinders near the transition into the SVBC phase.

Figure 4

Weak PVB order at $J_2=0.3$ on (a) $YC$7-3, (b) $XC$9-1, and (c) $XC$12-0 cylinders. The widths of the lines are proportional to $|⟨S_i·S_j⟩+0.32|$ for all the plots, with solid blue (dashed red) lines for negative (positive) values. The bond strength scale is indicated at the top. The three different sublattices of plaquettes are labeled as $A$, $B$, and $C$. In these figures, we keep $m=6000$ states in our DMRG calculation. The truncation error for $XC$9-1 is smaller than ${10}^{-7}$, while it is near ${10}^{-6}$ for the wider $YC$7-3 and $XC$12-0 cylinders.

Figure 5

PVB order correlation length $\xi$ for different cylinder circumferences $C$ at $J_2=0.3$. The black circles indicate $YC$ cylinders, while the red diamonds indicate $XC$ cylinders. The straight green line illustrates the expected linear behavior at the quantum critical point of the two-dimensional system. The inset shows the local PVB order parameter versus distance from the end of the cylinder in lattice spacings. The estimated PVB correlation length ${\xi }_p$ is indicated for each cylinder.