Competition between spin liquids and valence-bond order in the frustrated spin-$\frac{1}{2}$ Heisenberg model on the honeycomb lattice

Using variational wave functions and Monte Carlo techniques, we study the antiferromagnetic Heisenberg model with first-neighbor $J_1$ and second-neighbor $J_2$ antiferromagnetic couplings on the honeycomb lattice. We perform a systematic comparison of magnetically ordered and nonmagnetic states (spin liquids and valence-bond solids) to obtain the ground-state phase diagram. Néel order is stabilized for small values of the frustrating second-neighbor coupling. Increasing the ratio $J_2/J_1$, we find strong evidence for a continuous transition to a nonmagnetic phase at $J_2/J_1\approx 0.23$. Close to the transition point, the Gutzwiller-projected uniform resonating valence-bond state gives an excellent approximation to the exact ground-state energy. For $0.23\lesssim J_2/J_1\lesssim 0.36$, a gapless $Z_2$ spin liquid with Dirac nodes competes with a plaquette valence-bond solid. In contrast, the gapped spin liquid considered in previous works has significantly higher variational energy. Although the plaquette valence-bond order is expected to be present as soon as the Néel order melts, this ordered state becomes clearly favored only for $J_2/J_1\gtrsim 0.3$. Finally, for $0.36\lesssim J_2/J_1\le 0.5$, a valence-bond solid with columnar order takes over as the ground state, being also lower in energy than the magnetic state with collinear order. We perform a detailed finite-size scaling and standard data collapse analysis, and we discuss the possibility of a deconfined quantum critical point separating the Néel antiferromagnet from the plaquette valence-bond solid.

Figure 1

Phase diagram of the spin-$\frac{1}{2}{J}_1\text{−}{J}_2$ Heisenberg model on the honeycomb lattice for $0\le J_2/J_1\le 0.5$ with schematic illustrations of the Néel magnetic order, plaquette, and columnar dimer orders. The solid dots indicate quantum phase transitions between Néel and plaquette VBS ($J_2/J_1\approx 0.23$) and between plaquette and columnar VBS ($J_2/J_1\approx 0.36$). The region where the $d\pm id$ spin liquid has a competitive energy is marked by the green oval.

Figure 2

The honeycomb lattice is shown on the left: ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$ are the primitive vectors of the Bravais lattice. $A$ and $B$ denote the two sublattices: $A$-type sites are placed at the origin of the unit cell, while $B$-type sites are displaced by $(0,1)$. The dashed lines represent the directions of the vectors ${\mathbf{T}}_1$ and ${\mathbf{T}}_2$ that define the finite lattice clusters used in the calculations. A schematic illustration of the interactions in the $J_1\text{−}{J}_2$ Heisenberg model is shown on the right.

Figure 3

Schematic illustration of the $d\pm id$ spin-liquid state. Here, ${\varphi }_{ij}$ and ${\theta }_{ij}$ are the complex phases of first-neighbor hopping and second-neighbor pairing, respectively. The direction of the arrows ($i\to j$) indicates the convention of phases for the hopping terms.

Figure 4

Mean-field spectrum of the gapless $d\pm id$ pairing state. The energy bands are shown along the path $\mathrm{\Gamma }\to K\to M\to \mathrm{\Gamma }$ in the Brillouin zone (inset). The value of the second-neighbor pairing is $\mathrm{\Delta }=0.35t$, which is very close to the optimal value obtained for $J_2/J_1=0.35$. The bands show Dirac cones located at $\mathrm{\Gamma }$ and $K$ points. Note that the two bands are twice degenerate along $M\to \mathrm{\Gamma }$.

Figure 5

Patterns of the first-neighbor hoppings in the quadratic Hamiltonian (6) as found (a) in the plaquette VBS and (b) in the columnar VBS.

Figure 6

Accuracy of the variational energy for different wave functions on the 24-site cluster. Here, $\mathrm{\Delta }E$ is the difference between the variational ($E_{\mathrm{var}}$) and the exact ($E_{\mathrm{ex}}$) ground-state energy.

Figure 7

Finite-size scaling of the square magnetization $m^2$, Eq. (7), for different values of $J_2/J_1$. For $J_2=0$ the optimal value of the fictitious magnetic field is $h/t\approx 0.32$.

Figure 8

Thermodynamic limit of the magnetization $m$ as a function of $J_2/J_1$. The result from quantum Monte Carlo of Ref. [47] for $J_2=0$ is shown for comparison (red cross). The classical value is $m=0.5$.

Figure 9

Finite-size scaling analysis of the variational energy of different wave functions for $J_2/J_1=0.3$, 0.35, and 0.4. Statistical errors are smaller than the symbol size. The uRVB, SPS, and $d\pm id$ states are reported only for $J_2/J_1=0.3$ and 0.35; for $J_2/J_1=0.4$, their energies are much higher than those of plaquette and columnar dimer VBS states. The dashed lines are the fitting functions used for the extrapolation to the thermodynamic limit. For the $d\pm id$ state, the optimal values of the parameters are $\mathrm{\Delta }/t\approx 0.31$ and $\mathrm{\Delta }/t\approx 0.36$ for $J_2/J_1=0.3$ and 0.35, respectively. For the plaquette state the parameters range from $\mathrm{\Delta }/t\approx 0.31$ and $t^{\prime }/t\approx 0.90$ for $J_2/J_1=0.3$ to $\mathrm{\Delta }/t\approx 0.38$ and $t^{\prime }/t\approx 0.62$ for $J_2/J_1=0.4$. Finally, for the columnar state we get $\mathrm{\Delta }/t\approx 0.34$ ($\mathrm{\Delta }/t\approx 0.37$) and $t^{\prime }/t\approx 0.45$ ($t^{\prime }/t\approx 0.20$) for $J_2/J_1=0.35$ ($J_2/J_1=0.4$).

Figure 10

Finite-size scaling analysis of the variational energy of VBS and collinear magnetic states at $J_2/J_1=0.5$. The optimal values for the plaquette state are $\mathrm{\Delta }/t\approx 0.40$ and $t^{\prime }/t\approx 0.20$. For the columnar state $t^{\prime }/t$ is essentially zero, while $\mathrm{\Delta }/t\approx 0.37$. The fictitious magnetic field of the collinear state has optimal value $h/t\approx 0.98$.

Figure 11

Finite-size scaling collapse of the data of the antiferromagnetic (top) and plaquette (bottom) order parameters.

Figure 12

Symmetry generators of the point group of the honeycomb lattice ($\sigma$ and $R$). As an example, we show the loops for which we computed the $\text{SU}\left(2\right)$ flux: (a) parallelogram-shaped (sublattice $A$), (b) parallelogram-shaped (sublattice $B$), (c) diamond, and (d) rectangular plaquettes.