Quantum phase diagram of the spin-1 $J_1-J_2$ Heisenberg model on the honeycomb lattice

Strongly correlated systems with geometric frustrations can host the emergent phases of matter with unconventional properties. Here, we study the spin $S=1$ Heisenberg model on the honeycomb lattice with the antiferromagnetic first- ($J_1$) and second-neighbor ($J_2$) interactions ($0.0\le J_2/J_1\le 0.5$) by means of density-matrix renormalization group. In the parameter regime $J_2/J_1\lesssim 0.27$, the system sustains a Néel antiferromagnetic phase. At the large $J_2$ side $J_2/J_1\gtrsim 0.32$, a stripe antiferromagnetic phase is found. Between the two magnetic ordered phases $0.27\lesssim J_2/J_1\lesssim 0.32$, we find a nonmagnetic intermediate region with a plaquette valence-bond order. Although our calculations are limited within a six unit-cell width on cylinder, we present evidence that this plaquette state could be a strong candidate for this nonmagnetic region in the thermodynamic limit. We also briefly discuss the nature of the quantum phase transitions in the system. We gain further insight into the nonmagnetic phases in the spin-1 system by comparing its phase diagram with the spin-$1/2$ system.

Figure 1

Quantum phase diagram of the spin-$1J_1-J_2$ Heisenberg model on the honeycomb lattice. With increasing $J_2$ coupling, the system has a Néel AFM phase for $J_2<0.26$, a stripe AFM phase for $J_2\gtrsim 0.32$, and an intermediate non-magnetic phase with a plaquette valence-bond order in our DMRG calculations. As the finite-size effects on different cylinder geometries, the first transition point is estimated at $0.26\sim 0.28$.

Figure 2

Cylinder geometries used in the DMRG calculations. (a) is an AC4-6 cylinder and (b) is a ZC4-6 cylinder.

Figure 3

Spin correlation functions in real space for $J_2=0.0$ on the (a) AC4-24 cylinder and (b) ZC4-24 cylinder. The green site is the reference site in the middle of the cylinder, and the blue solid and red shaded circles denote the positive and negative correlations, respectively. The radius of the circle is proportional to the magnitude of correlations. Here, we only show the left half $2\times 4\times 12$ sites.

Figure 4

(a) and (b) are the log-linear plots of spin correlations on the AC4-24 and ZC4-24 cylinders. (c) and (d) are the $J_2$ dependence of the long-distance spin correlations $S_d\equiv \sqrt{|\langle S_0{S}_d\rangle |}$ [$d$ is the longest distance in (a) and (b)]. In subfigure (a), the correlation data for $J_2<0.27$ are all positive. For $J_2=0.27$, the Néel AFM pattern is destructed, thus the correlations change sign in some places.

Figure 5

Spin correlation function $\langle {\vec{S}}_0·{\vec{S}}_j\rangle$ in real space for $J_2=0.4$ on the (a) AC6-18 cylinder and (b) ZC6-18 cylinder. The green site is the reference spin ${\vec{S}}_0$ in the middle of the lattice, and the blue solid and red shaded circles denote the positive and negative spin correlations, respectively. The area of the circle is proportional to the amplitude of correlations. The dashed rectangles denote the unit cells.

Figure 6

Log-linear plots of spin correlations at large $J_2$ side for the (a) AC4-24 and (b) ZC4-24 cylinders. (c) Cylinder width dependence of the bulk ground-state energy for $J_2=0.4$ on the AC and ZC cylinders. ZC cylinders always have a lower energy than AC cylinders.

Figure 7

The nearest-neighbor bond energy for $J_2=0.5$ on the ZC4-24 cylinder. Here, we only show the left half lattice. The positive horizontal (labeled by blue) and the negative vertical (labeled by red) bond energies are consistent with the spin configuration in the stripe AFM state.

Figure 8

The nearest-neighbor bond energy textures for $J_2=0.3$ on the (a) AC6-24 cylinder and (b) tZC6-18 cylinder, which are obtained by keeping $8000SU\left(2\right)$ states. The bond textures are obtained by subtracting the average bond energy. The blue and red bonds denote the positive and negative bond textures, respectively. On both geometries, a PVB bond pattern is found. For AC6-24 in (a), only the left half lattice is shown here.

Figure 9

(a) $J_2$ dependence of the spin gap on the ZC4 and ZC6 cylinder systems. (b) Cylinder width dependence of the bulk ground-state energy on AC (AC4 and AC6) and ZC (ZC4 and ZC6) cylinders for $J_2=0.3$.

Figure 10

(a) $J_2$ dependence of the bipartite entanglement entropy on the ZC4 cylinder. At $J_2\simeq 0.28$ and $0.32$, the entropy has the sharp jump and drop, which appear consistent with the quantum phase transitions. Entanglement spectra for (b) $J_2=0.0$ on the ZC6-18 cylinder, (c) $J_2=0.4$ on the ZC6-18 cylinder, and (d) $J_2=0.3$ on the ZC4-24 cylinder. The blue numbers denote the number of the largest eigenvalues in each $S$ sector. The green dashed lines denote the tower of states structure in the magnetic ordered states.