Extent of frustration in the classical Kitaev-$\mathrm{\Gamma }$ model via bond anisotropy

In the pseudospin-$\frac{1}{2}$ honeycomb Mott insulators with strong spin-orbit coupling, there are two types of bond-dependent exchange interactions, named Kitaev $\left(K\right)$ and $\mathrm{\Gamma }$, leading to strong frustration. While the ground state of the Kitaev model is a quantum spin liquid with fractionalized excitations, the ground state of the $\mathrm{\Gamma }$ model remains controversial. In particular, the phase diagram of the $K\mathrm{\Gamma }$ model with ferromagnetic $K$ and antiferromagnetic $\mathrm{\Gamma }$ interactions has been intensively studied because of its relevance to candidate materials such as $\alpha \text{−}{\mathrm{RuCl}}_3$. Numerical studies also included the effects of tuning the bond strengths, i.e., $z$-bond strength different from the other bonds. However, no clear consensus on the overall phase diagram has been reached yet. Here we study the classical $K\mathrm{\Gamma }$ model with anisotropic bond strengths using Monte Carlo simulations to understand the phases that emerge out of competition between the two frustrated limits. We also address how the anisotropic bond strength affects the phase diagram and strength of quantum fluctuations. We found various large unit cell phases due to the competing frustrations, and analyzed their intrinsic degeneracy based on the symmetry of the Hamiltonian. Using the linear spin wave theory we showed that the anisotropic bond strength enhances quantum fluctuations in the $\mathrm{\Gamma }$-dominant regime where a small reduced moment is observed. The implications of our findings in relation to the quantum model are also discussed.

Figure 1

The honeycomb lattice in the strong $z$-bond limit $(g>0)$ where the enhanced $z$-bond interaction strength is indicated in red and the $xy$ chain extends horizontally. The three plaquette sublattices $\overline{A},\overline{B},\overline{C}$ are highlighted in dark gray, light gray, and white, respectively. The cubic $xyz$ and crystallographic $XYZ$ bases are also shown, where $\widehat{Z}$ is perpendicular to the honeycomb plane and $\widehat{X},\widehat{Y}$ are perpendicular and parallel to the $z$-bond direction, respectively.

Figure 2

The classical phase diagram of the anisotropic $K\mathrm{\Gamma }$ model. All phases are separated by first-order transitions except for the ${18}^{\eta }$ and ${18}^{\prime }$ phase boundary, which is of second-order. We highlight three points with macroscopic degeneracy, the $K=\mathrm{\Gamma }$ dimer point and the pure isotropic $K$ and $\mathrm{\Gamma }$ points, with a black diamond, triangle, and circle, respectively. In the isotropic limit the three orientations of $\text{ZZ}+12$, $6+18$, and $16+48$ phases are degenerate, which we indicate by the dark red, dark blue, and dark purple lines along $g=0$, respectively. This degeneracy is lifted for $g\ne 0$ due to the broken $C_3$ symmetry. At $\psi =0$ we indicate the Kitaev FM dimer phase in dark gray, which immediately forms the $\text{FM}+{\text{120}}^{\circ }$ order in the presence of $\mathrm{\Gamma }>0$. See the main text for the ordering patterns of the phases and their SSF peaks in Appendix pp2.

Figure 3

The (a) ${\text{ZZ}}_z$ and (b) ${\text{12}}_z$ configurations at $(\psi ,g)=(0.4\pi ,0.94)$. The plaquette sublattice $\overline{A}$ is shown in gray. The color of each moment in Figs. 3–5 denotes the angle made with respect to the $\left[111\right]$ direction.

Figure 4

The (a) $6_z$ and (b) ${18}_z$ configurations at $(\psi ,g)=(0.25\pi ,0.84)$, where the counterrotating spiral pattern ABCCBA pattern is shown. Whereas the $6_z$ orientation contains a chain of FM aligned moments along the $x$ and $y$ bonds, in the $6_y$ orientation the chain runs along the $x$ and $z$ bonds.

Figure 5

The (a) ${18}_V^{\eta }$, (b) ${18}_z^{\eta }$, (c) ${18}_x^{\eta }$, and (d) ${18}_y^{\eta }$ configurations at $(\psi ,g)=(0.4\pi ,0.5)$.

Figure 6

$\eta$ representation of the moments ${\vec{S}}_i$ in the pure $\mathrm{\Gamma }$ limit, where we only display the ${\eta }_p$ that reside on the plaquette sublattice $\overline{A}$ (shown in gray) for clarity.

Figure 7

The (a) ${18}^{\eta }$ order at $(\psi ,g)=(0.4\pi ,0.5)$ and (b) ${18}^{\prime }$ order at $(\psi ,g)=(0.4\pi ,0.75)$, with the values of the plaquette flux $W_p/S^6$ shown in the center of each hexagon. The nine-plaquette unit cell is indicated by dashed lines. Note that in (b) the $W_p$ are inversion symmetric about the zero-flux plaquette as the ${18}^{\prime }$ phase respects inversion symmetry.

Figure 8

Reduced moment $\langle M\rangle /S$ (red) and magnon gap ${\mathrm{\Delta }}_0$ (blue) as a function of $g$ for (a) $\psi =0.2\pi$ and (b) $\psi =0.4\pi$. The classical phases stabilized in each region are labeled and the phase boundaries are indicated by the dashed black lines.

Figure 9

The two unit cells used to construct the Monte Carlo cluster, with vectors ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$ shown along with the three bond types. From left to right: the rhombic unit cell with unit cell vectors Eq. (A1) and the rectangular unit cell with unit cell vectors Eq. (A2). The bond type $x,y,z$ are colored green, blue, and red, respectively.

Figure 10

The accessible momentum points for the $N=4\times 12\times 6=288$ site cluster constructed using the rectangular unit cell in Eq. (A2). The red and green hexagons correspond to the first and second Brillouin zones, respectively. The three $M$ points are shown and we use the label $x,y,z$ to distinguish different orientations of the same magnetic order: see Appendix pp2.

Figure 11

SSF of the (a) ${\text{ZZ}}_z$ and (b) ${12}_z$ patterns at $(\psi ,g)=(0.4\pi ,0.94)$, as well as the (c) ${\text{ZZ}}_y$ and (d) ${12}_y$ patterns at $(0.15\pi ,0.94)$ and $(0.1\pi ,0.65)$, respectively. The ZZ peaks are located at one of the three $M$ points, whereas the 12-site peaks are located at one of three $\frac{1}{2}K,\frac{1}{2}{K}^{\prime }$ pairs. The color of each ordering wave vector in Figs. 11–16 denotes the relative intensity of each peak.

Figure 12

SSF of the (a) $6_z$ and (b) ${18}_z$ patterns at $(0.25\pi ,0.84)$, as well as the (c) $6_y$ and (d) ${18}_y$ patterns at $(0.35\pi ,0.84)$. The six-site peaks are located at one of three $\frac{2}{3}M,\frac{4}{3}M$ pairs, whereas the 18-site peaks are located at multiple $\frac{2}{3}M,\frac{4}{3}M$ pairs.

Figure 13

SSF of the (a) ${18}_V^{\eta }$ and (b) ${18}_y^{\eta }$ patterns at $(0.4\pi ,0.5)$, as well as the (c) ${18}_{z_1}^{\prime }$ and (d) ${18}_y^{\prime }$ phase at $(0.4\pi ,0.71)$. The ordering vectors are located at multiple $\frac{2}{3}M,\frac{4}{3}M$ pairs.

Figure 14

SSF of the (a) ${\text{FM}}_z$ and (b) ${120}_z^{\circ }$ phase at $(0.04\pi ,0.5)$. The dominant SSF peak is at the $\mathrm{\Gamma }$ point for the FM phase and one of three $K,K^{\prime }$ pairs for the ${120}^{\circ }$ phase.

Figure 15

SSF of the (a) ${24}_z$ and (b) ${24}_y$ patterns at $(0.125\pi ,0.57)$. The ${24}_z$ dominant SSF peaks are at $\frac{1}{2}{M}_z$ whereas for ${24}_y$ they are at $(\frac{1}{12},\frac{5}{12}),(-\frac{1}{12},\frac{7}{12})$ in the $\{{\mathbf{b}}_1,{\mathbf{b}}_2\}$ basis, where ${\mathbf{b}}_{1,2}$ satisfy ${\mathbf{a}}_i·{\mathbf{b}}_j=2\pi {\delta }_{ij}$.

Figure 16

SSF of the (a) ${16}_y$ and (b) ${48}_y$ patterns at $(0.14\pi ,0.5)$. The ${16}_y$ dominant SSF peaks are at $\frac{3}{4}{M}_y,\frac{5}{4}{M}_y,$ whereas for ${48}_y$ they are at $(-\frac{1}{3},-\frac{1}{24}),(\frac{1}{3},\frac{7}{24})$ in the $\{{\mathbf{b}}_1,{\mathbf{b}}_2\}$ basis, where ${\mathbf{b}}_{1,2}$ satisfy ${\mathbf{a}}_i·{\mathbf{b}}_j=2\pi {\delta }_{ij}$.