One Proximate Kitaev Spin Liquid in the $K\text{−}J\text{−}\mathrm{\Gamma }$ Model on the Honeycomb Lattice

In addition to the Kitaev ($K$) interaction, candidate Kitaev materials also possess Heisenberg ($J$) and off-diagonal symmetric ($\mathrm{\Gamma }$) couplings. We investigate the quantum ($S=1/2$) $K\text{−}J\text{−}\mathrm{\Gamma }$ model on the honeycomb lattice by a variational Monte Carlo method. In addition to the “generic” Kitaev spin liquid (KSL), we find that there is just one proximate KSL (PKSL) phase, while the rest of the phase diagram contains different magnetically ordered states. The PKSL is a gapless $Z_2$ state with 14 Majorana cones, which in contrast to the KSL has a gapless spin response. In a magnetic field applied normal to the honeycomb plane, it realizes two of Kitaev’s gapped chiral spin-liquid phases, of which one is non-Abelian with Chern number $\nu =5$ and the other is Abelian with $\nu =4$. These two phases could be distinguished by their thermal Hall conductance.

Figure 1

(a) Phase diagram of the quantum $K\text{−}J\text{−}\mathrm{\Gamma }$ model for $K<0$ in the limit of large system size. There are two QSL phases of different types but with the same PSG, the GKSL and the PKSL. The magnetically ordered phases are antiferromagnetic (AFM), stripe, incommensurate spiral (IS), zigzag, and ferromagnetic (FM) order. (b) Detail of the limit $|K|/\mathrm{\Gamma }\to 0$, where all phases are magnetically ordered; the transitions occur at $J/\mathrm{\Gamma }=0.15$, 0.05, and $-0.75$.

Figure 2

(a) Spinon dispersion in the GKSL, drawn with $\mathrm{\Gamma }/|K|=0.1$ and $J=0$, showing 2 Majorana cones. (b) Spinon dispersion in the PKSL, drawn with $\mathrm{\Gamma }/|K|=0.3$ and $J=0$, showing 14 Majorana cones. (c) Ground-state energy per site of the $K\text{−}J\text{−}\mathrm{\Gamma }$ model at fixed $J/|K|=0.05$, showing a clear first-order phase transition. (d) Locations of the 14 cones of the PKSL in the first Brillouin zone. (e) Dynamical structure factor of the PKSL at low energy (integrated over energies $0\le \omega /|K|\le 0.08$). (f) Dynamical structure factor of the PKSL at $\omega /|K|=0.15$ (integrated over $0.13\le \omega /|K|\le 0.17$).

Figure 3

Phase diagrams of the PKSL ($\mathrm{\Gamma }/|K|=0.3$, $J=0$) in a magnetic field applied in the $\widehat{x}-\widehat{y}$ direction and in the $\widehat{c}=(\widehat{x}+\widehat{y}+\widehat{z})/\sqrt{3}$ direction. “6-cone” denotes a phase whose low-energy spinon dispersion has six remaining cones.