Density-matrix renormalization group study of the extended Kitaev-Heisenberg model

We study an extended Kitaev-Heisenberg model including additional anisotropic couplings by using the two-dimensional density-matrix renormalization group method. Calculating the ground-state energy, entanglement entropy, and spin-spin correlation functions, we make a phase diagram of the extended Kitaev-Heisenberg model around the spin-liquid phase. We find a zigzag antiferromagnetic phase, a ferromagnetic phase, a ${120}^{\circ }$ antiferromagnetic phase, and two kinds of incommensurate phases around the Kitaev spin-liquid phase. Furthermore, we study the entanglement spectrum of the model, and we find that entanglement levels in the Kitaev spin-liquid phase are degenerate forming pairs, but those in the magnetically ordered phases are nondegenerate. The Schmidt gap defined as the energy difference between the lowest two levels changes at the phase boundary adjacent to the Kitaev spin-liquid phase. However, we find that phase boundaries between magnetically ordered phases do not necessarily agree with the change of the Schmidt gap.

Figure 1

Honeycomb lattice with $6\times 8$ sites with the periodic boundary condition. Blue dotted, red dashed-dotted, and green solid bonds labeled by ${\mathcal{J}}_x,{\mathcal{J}}_y$, and ${\mathcal{J}}_z$ have $S^x{S}^x,S^y{S}^y$, and $S^z{S}^z$ terms in a Kitaev model, respectively. We define the $x$-axis direction as an armchair-edge direction and the $y$-axis direction as a zigzag-edge direction.

Figure 2

Phase diagram of the extended KH model (1). There are a ferromagnetic phase (FM), a ${120}^{\circ }$ AFM phase $\left({120}^{\circ }\right)$, a Kitaev spin-liquid phase (SL), two incommensurate phases (IC1, IC2), and a zigzag-type antiferromagnetic phase (zigzag). The circle and $\times$ points are determined by the second derivative of energy with respect to $I_2$ and connected by lines. The boundaries denoted by blue solid lines are expected to be of first-order transition, and those by green broken lines to be of continuous transition.

Figure 3

(a) $\langle S_i^x{S}_j^x\rangle$, (b) $\langle S_i^y{S}_j^y\rangle$, and (c) $\langle S_i^z{S}_j^z\rangle$ for zigzag-type AFM phase at $I_1/J=3.8$ and $I_2/J=-3.8$. The $i$ site is indicated by a brown rhombus point. Upward red arrows and downward blue arrows show positive and negative values of spin-spin correlation, respectively. The length of the arrows represents the strength of spin-spin correlation.

Figure 4

(a) The ground-state energy per site, $E$ (red plots), (b) second derivative of $E$ with respect to $I_2,d^{2}E/d{I}_2^2$ (green plots), and (c) entanglement entropy (blue plots). $I_1/J=0.63$. The vertical dotted lines denote the phase boundary determined by the second derivative of $E$.

Figure 5

(a) $\langle S_i^x{S}_j^x\rangle$, (b) $\langle S_i^y{S}_j^y\rangle$, and (c) $\langle S_i^z{S}_j^z\rangle$ for the IC2 phase at $I_1/J=2.5$ and $I_2/J=5.0$. The $i$ site is indicated by a brown rhombus point. Upward red arrows and downward blue arrows show positive and negative values of spin-spin correlation, respectively. The length of the arrows represents the strength of spin-spin correlation.

Figure 6

Entanglement spectrum for a zigzag-type AFM ordered ground state at $I_1/J=3.8$ and $I_2/J=-3.8$.

Figure 7

Entanglement spectrum for the extended KH model (1). $I_1/J=0.63$. Blue crosses represent entanglement levels and gray lines connect the spectrum belonging to the same entanglement levels. $\left[n\right]$ denotes $n$-fold degeneracy of the lowest entanglement level in each phase.

Figure 8

Same as Fig. 4, but $I_1/J=3.8$.

Figure 9

Same as Fig. 5, but for the IC1 phase at $I_1/J=2.5$ and $I_2/J=2.5$.

Figure 10

Same as Fig. 7, but $I_1/J=3.8$.

Figure 11

Same as Fig. 4, but $I_1/J=-1.3$.

Figure 12

Same as Fig. 7, but $I_1/J=-1.3$.

Figure 13

The entanglement spectrum for the KH model (A1). Blue crosses represent entanglement levels and gray lines connect the spectrum belonging to the same entanglement levels. $\left[n\right]$ denotes $n$-fold degeneracy of the lowest entanglement level in each phase.

Figure 14

Cluster configuration of $6\times 8$ sites. The numbers label sites on a honeycomb lattice. Vertical dashed lines denote the cutting position when we divide the whole system into two subsystems, $A$ and $B$. For the cylindrical boundary condition the system is divided only once at the middle vertical line, while for the toroidal boundary condition the system is cut twice at the middle vertical line and the right or left vertical line. $W_1$ and $W_2$ show the Wilson loops defined on a hexagon on a honeycomb lattice, which crosses the middle vertical line. $W_3$ and $W_4$ show the loops that cross the right or left vertical line.

Figure 15

The dependence of the entanglement spectrum ${\xi }_i$ of the Kitaev spin liquid on system size and boundary condition. (a) $6\times 8$-site system (blue rhombuses) and $6\times 20$-site system (red circles) with the cylindrical boundary condition, (b) $6\times 8$-site system (blue rhombuses) and $6\times 30$-site system (red circles) with the toroidal boundary condition, (c) $10\times 8$-site system (blue rhombuses) and $10\times 20$-site system (red circles) with the cylindrical boundary condition, and (d)$10\times 8$-site system (blue rhombuses) and $10\times 20$-site system (red circles) with the toroidal boundary condition.