Understanding thermal and quantum fluctuations in extended Kitaev-Yao-Lee spin-orbital model

Abstract

Building upon the spin-1/2 Kitaev model on a honeycomb lattice, the Yao-Lee spin-orbital model provides exactly solvable quantum spin liquids with potentially better stability against perturbations due to the additional degree of freedom. Recently, the microscopic mechanism underlying the Yao-Lee interaction in honeycomb materials has been uncovered, leading to an extended Kitaev-Yao-Lee spin-orbital model when the celebrated Kugel-Khomskii interaction is included. Numerical studies of this model have identified various disordered phases, including a broad region of the nematic phase that is reminiscent of a spin-orbital liquid. Here, we investigate the origin and stability of this nematic phase via thermal and quantum fluctuations using classical Monte Carlo simulations and a generalized spin wave theory appropriate for the spin-orbital model. We demonstrate that the additional spin-orbital degree of freedom gives rise to strong thermal and quantum fluctuations in spin-orbital models, providing insight into the emergence of disordered phases.

Figures (4)

• Figure 1: Classical phase diagrams of the spin-orbital model at various temperatures. The unit of temperature is $\mathrm{\tilde{K}}=k_BT/J$. (a) The stripy spin (SS) and the antiferromagnetic orbital (AFO) configurations of the ${\rm SS \times AFO}$ phase. (b) At zero temperature, when $a>0$ and $b>0$, the model is fully ordered (${\rm SS \times AFO}$) with the stripy spin and the antiferromagnetic orbital (white region). The YL point is $a=0$ and $b=0$ (gray circle). The Kitaev phase (K) is the green line $a=0$ and $b>0$. The nematic phase (NP) is the red line $a>0$ and $b=0$. (c) At a low temperature $T=0.001 \,\mathrm{\tilde{K}}$, the Kitaev phase and the YL point remain unchanged, but the NP extends to $b>0$ (red region). The red and green stars highlight the parameters used to study the finite-temperature phase transitions in Fig. \ref{['Fig3']}. (d) At a higher intermediate temperature $T=0.005 \,\mathrm{\tilde{K}}$, the Kitaev phase, the YL point, and the NP are extended by thermal fluctuations.
• Figure 2: Example of a Cartesian state Baskaran_PRB2008 of the classical Kitaev model on a $4\times4$ lattice. The two neighbouring spins are aligned along the $x$, $y$, and $z$ axes on the red, green, and blue bonds, respectively, meaning the minimum bond energy is satisfied on these bonds. For the remaining black bonds, the energy is zero. Each unit cell has one satisfied bond.
• Figure 3: Structure factors of spin $S(\boldsymbol{q=M})$, orbital $T(\boldsymbol{q=0})$, and spin-orbital $ST(\boldsymbol{q=M})$ vs. temperature. The SS$\times$AFO ground state has both maximum $S(\boldsymbol{M})$, $T(\boldsymbol{0})$, and $ST(\boldsymbol{M})$. (a) When $a=0.1$ and $b=0.05$, denoted by the green star in Fig. \ref{['Fig1']}(c), the Kitaev phase emerges at intermediate temperatures, characterized by the maximum $T(\boldsymbol{0})$ only. (c) When $a=0.5$ and $b=0.01$, denoted by the red star in Fig. \ref{['Fig1']}(c), the NP emerges at low temperatures, characterized by the maximum $ST(\boldsymbol{M})$ and a small $S(\boldsymbol{M})$. The transition from the NP to the ${\rm SS \times AFO}$ is broad and sensitive to finite-size effects due to the energy cost of the NP. See the main text for discussion.
• Figure 4: Quantum phase diagram of the spin-orbital model by a generalized spin wave theory at various expansion orders (the generalized spin length $\Lambda$). The theory computes the phase boundary using the magnon gap-closing condition of the fully ordered ${\rm SS \times AFO}$ phase. The Kitaev phase boundary (green line order $\Lambda^{-1}$) has the vanishing spin magnon, while the NP boundary (red solid line for order $\Lambda^{-1}$ and dashed line for order $\Lambda^0$) has the vanishing spin-orbital magnon.