Negative sign free formulations of generalized Kitaev models with higher symmetries

We provide a negative sign free formulation of the auxiliary field quantum Monte Carlo algorithm for generalized Kitaev models with higher symmetries. Our formulation is based on the Abrikosov fermion representation of the spin-1/2 degree of freedom and the phase pinning approach [T. Sato and F. F. Assaad, Phys. Rev. B 104, L081106 (2021)]. Enhancing the number of fermion flavors or orbitals from 1 to $N$ allows one to generalize the inherent $Z_2$ global symmetry to $Z_2\times \mathrm{S}\mathrm{U}{\left(N\right)}_o$. Using this general approach, we study the $Z_2\times \mathrm{S}\mathrm{U}{\left(2\right)}_o$ Kitaev-Heisenberg model reflecting the competition between the isotropic Heisenberg exchange and Kitaev-type bond-directional exchange interactions. We show that the symmetry enhancement provides a path to escape frustration and that the spin liquid phases in the original $Z_2$-symmetric model are not present in this model. Nevertheless, the ground-state phase diagram is extremely rich and has points with higher global and local continuous symmetries as well as deconfined quantum critical points.

Figure 1

The ground-state phase diagram of the $Z_2\times \mathrm{S}\mathrm{U}{\left(N\right)}_o$ Kitaev-Heisenberg model as a function of the angle $\phi$.

Figure 2

Real-space correlations of ${\widehat{S}}^1$ [(a), (b)] and $\widehat{\mathit{T}}$ [(c), (d)] and dynamical structure factor of $\widehat{\mathit{T}}$ generators, $C^T(\mathit{q},\omega )$ [(e), (f)] in the Kitaev limits at $N=2$. (a), (c), (e) $\phi /\pi =0.5$ (AFM Kitaev limit) and (b), (d), (f) $\phi /\pi =1.5$ (FM Kitaev limit). (g) First (solid) and second (dashed line) Brillouin zones. Results in (e), (f) correspond to scans along the red line. Here, $L=9$ and $T=1/200$.

Figure 3

Real-space correlations of ${\widehat{S}}^1$ (top panel), $\widehat{\mathit{T}}$ (middle panel), and ${\widehat{D}}^T$ (bottom panel) at different values of the angle $\phi$. (a), (b), (d), (e) $T=1/80$ and (c), (f) $T=1/200$. Here, $L=9$ and $N=2$.

Figure 4

(a) Correlation ratios for AFM (upper panel) and KVBS (lower panel) states at different values of $\phi /\pi$ close to the SU(4)-symmetric AFM Heisenberg point at $\phi /\pi =0$. (b) Correlation ratio for a FM state at different values of $\phi /\pi$ close to the SU(4)-symmetric FM Heisenberg point at $\phi /\pi =1.0$. Here, $T=1/80$.

Figure 5

Dynamical structure factor of $\widehat{\mathit{T}}$ generators, $C^T(\mathit{q},\omega )$, at different values of the angle $\phi$. (a), (b), (c), (e), (f), (g) $T=1/80$ and (d), (h) $=1/200$. Results correspond to scans along the red line of first and second Brillouin zones in Fig. 2. Here, $L=9$ and $N=2$.