Quantum Monte Carlo simulation of generalized Kitaev models

Frustrated spin systems generically suffer from the negative sign problem inherent in Monte Carlo methods. Since the severity of this problem is formulation dependent, optimization strategies can be put forward. We introduce a phase pinning approach in the realm of the auxiliary field quantum Monte Carlo algorithm. If we can find an antiunitary operator that commutes with the one-body Hamiltonian coupled to the auxiliary field, then the phase of the action is pinned to 0 and $\pi$. For generalized Kitaev models, we can successfully apply this strategy and observe a remarkable improvement of the average sign. We use this method to study the thermodynamical and dynamical properties of the Kitaev-Heisenberg model down to temperatures corresponding to half of the exchange coupling constant. Our dynamical data reveal finite temperature properties of ordered and spin-liquid phases inherent in this model.

Figure 1

(a) Spin-1/2 degrees of freedom ${\widehat{\mathit{S}}}_{\mathit{i}}$ on the honeycomb lattice are subject to Heisenberg $J{\widehat{\mathit{S}}}_i·{\widehat{\mathit{S}}}_{i+\delta }$ and Kitaev $2K{\widehat{S}}_i^{\delta }{\widehat{S}}_{i+\delta }^{\delta }$ exchange interactions. Here, $\delta =1$ (red), 2 (green), and 3 (blue) runs over the bonds, and ${\mathit{a}}_1$ and ${\mathit{a}}_2$ correspond to the lattice vectors. (b) First (solid) and second (dashed line) Brillouin zones. (c) Average sign $\langle \text{sign}\rangle$ as a function of $V$ for various angles $\phi$. The figure includes the ground-state phase diagram with antiferromagnetic (AFM), Kitaev spin-liquid (KSL), zigzag (ZZ), ferromagnetic (FM), and stripy (SP) phases, as proposed in Ref. [3]. Here, we have set the temperature to $T=1$ in units of $A$.

Figure 2

$T$ dependence of inverse uniform spin susceptibilities $1/\chi$ at different values of $\phi /\pi$. The dashed line indicates the Curie's law considered here.

Figure 3

Real-space spin-spin correlations $\langle {\widehat{S}}_{\mathit{r}}^1{\widehat{S}}_{\mathit{0}}^1\rangle$ (top panel) and momentum resolved spin susceptibility $\chi \left(\mathit{q}\right)=\frac{1}{3}{\sum }_{\alpha }{\chi }_{\alpha }\left(\mathit{q}\right)$ (bottom panel) in the first (solid) and second (dashed line) Brillouin zones [see Fig. 1]. (a), (b) $\phi /\pi =0.8$ ($T=1/2.6$); (c), (d) $\phi /\pi =1.7$ ($T=1/1.9$); (e), (f) $\phi /\pi =0.5$ ($T=1/1.6$); and (g), (h) $\phi /\pi =1.5$ ($T=1/1.6$).

Figure 4

Dynamical spin structure factor $C(\mathit{q},\omega )$ at different values of $\phi /\pi$. Here, $T=1/1.6$. Results used here correspond to scans along the red line of Fig. 1.

Figure 5

Dynamical spin structure factor $C(\mathit{q},\omega )$ at $\phi /\pi =0.8$ for higher [(a) $T=1/0.5$] and lower [(b) $T=1/2.6$] temperatures.