We study quantum disordered ground states of the two-dimensional Heisenberg-Kitaev model on the triangular lattice using a Schwinger boson approach. Our aim is to identify and characterize potential gapped quantum spin liquid phases that are stabilized by anisotropic Kitaev interactions. For antiferromagnetic Heisenberg and Kitaev couplings and sufficiently small spin $S$ , we find three different symmetric ${Z}_{2}$ spin liquid phases, separated by two continuous quantum phase transitions. Interestingly, the gap of elementary excitations remains finite throughout the transitions. The first spin liquid phase corresponds to the well-known zero-flux state in the Heisenberg limit, which is stable with respect to small Kitaev couplings and develops ${120}^{\ensuremath{\circ}}$ order in the semiclassical limit at large $S$ . In the opposite Kitaev limit, we find a different spin liquid ground state, which is a quantum disordered version of a magnetically ordered state with antiferromagnetic chains, in accordance with results in the classical limit. Finally, at intermediate couplings, we find a spin liquid state with unusual spin correlations. Upon spinon condensation, this state develops Bragg peaks at incommensurate momenta in close analogy to the magnetically ordered Z2 vortex crystal phase, which has been analyzed in recent theoretical works.We study quantum disordered ground states of the two-dimensional Heisenberg-Kitaev model on the triangular lattice using a Schwinger boson approach. Our aim is to identify and characterize potential gapped quantum spin liquid phases that are stabilized by anisotropic Kitaev interactions. For antiferromagnetic Heisenberg and Kitaev couplings and sufficiently small spin $S$, we find three different symmetric ${Z}_{2}$ spin liquid phases, separated by two continuous quantum phase transitions. Interestingly, the gap of elementary excitations remains finite throughout the transitions. The first spin liquid phase corresponds to the well-known zero-flux state in the Heisenberg limit, which is stable with respect to small Kitaev couplings and develops ${120}^{\ensuremath{\circ}}$ order in the semiclassical limit at large $S$. In the opposite Kitaev limit, we find a different spin liquid ground state, which is a quantum disordered version of a magnetically ordered state with antiferromagnetic chains, in accordance with results in the classical limit. Finally, at intermediate couplings, we find a spin liquid state with unusual spin correlations. Upon spinon condensation, this state develops Bragg peaks at incommensurate momenta in close analogy to the magnetically ordered Z2 vortex crystal phase, which has been analyzed in recent theoretical works.

arXiv:1611.05454Physical review. B./Physical review. BJan 20, 201726 citations

Spin Liquid in the Triangular Lattice Heisenberg Model

Ian P. McCulloch, S. N. Saadatmand

Bulletin of the American Physical SocietyMar 14, 20160 citations

Chiral bosonic quantum spin liquid in the integer-spin Heisenberg-Kitaev model

Arnaud Ralko, Jaime Merino

Motivated by the possibility of finding a bosonic quantum spin liquid in the integer spin- $S$ Heisenberg-Kitaev model on the honeycomb lattice, we derive a Schwinger boson mean-field theory involving both singlet and triplet pairing channels which includes hopping and pairing operators on equal footing. The mixed construction introduced here is justified by the good comparison with exact diagonalization energies of the $S\ensuremath{\le}3/2$ Heisenberg-Kitaev model and the perfect match with the Luttinger-Tisza semiclassical energies obtained at large $S$ . We find various competing gapped quantum spin liquids close to the Kitaev point. A comparison of their spin excitation spectrum with the dynamical structure factor obtained from exact diagonalizations allows us to identify the physical spin liquid Ansatz of the model. In particular, we identify a chiral quantum spin liquid state whose spin excitation spectrum follows closely the exact diagonalization data and survives up to large spin $S\ensuremath{\lesssim}2$. We propose this state as a promising quantum spin liquid candidate for the integer spin- $S$ antiferromagnetic Kitaev model which may be realized in $S=1$ Kitaev materials ${\mathrm{A}}_{3}{\mathrm{Ni}}_{2}{\mathrm{XO}}_{6}$ and ${\mathrm{KNiAsO}}_{4}$ .Motivated by the possibility of finding a bosonic quantum spin liquid in the integer spin-$S$ Heisenberg-Kitaev model on the honeycomb lattice, we derive a Schwinger boson mean-field theory involving both singlet and triplet pairing channels which includes hopping and pairing operators on equal footing. The mixed construction introduced here is justified by the good comparison with exact diagonalization energies of the $S\ensuremath{\le}3/2$ Heisenberg-Kitaev model and the perfect match with the Luttinger-Tisza semiclassical energies obtained at large $S$. We find various competing gapped quantum spin liquids close to the Kitaev point. A comparison of their spin excitation spectrum with the dynamical structure factor obtained from exact diagonalizations allows us to identify the physical spin liquid Ansatz of the model. In particular, we identify a chiral quantum spin liquid state whose spin excitation spectrum follows closely the exact diagonalization data and survives up to large spin $S\ensuremath{\lesssim}2$. We propose this state as a promising quantum spin liquid candidate for the integer spin-$S$ antiferromagnetic Kitaev model which may be realized in $S=1$ Kitaev materials ${\mathrm{A}}_{3}{\mathrm{Ni}}_{2}{\mathrm{XO}}_{6}$ and ${\mathrm{KNiAsO}}_{4}$.

arXiv:2405.10731Physical review. B./Physical review. BOct 1, 20244 citations

Concept and realization of Kitaev quantum spin liquids

H. Takagi, T. Takayama, George Jackeli +2 more

arXiv:1903.08081Nature Reviews PhysicsMar 25, 2019813 citations

Comprehensive study of the global phase diagram of the J−K−Γ model on a triangular lattice

Shi Wang, Zhongyuan Qi, Bin Xi +3 more

The celebrated Kitaev honeycomb model provides an analytically tractable example with an exact quantum spin liquid ground state. While in real materials, other types of interactions besides the Kitaev coupling ( $K$ ) are present, such as Heisenberg ( $J$ ) and symmetric off-diagonal ($\mathrm{\ensuremath{\Gamma}}$) terms, these interactions can also be generalized to a triangular lattice. Here, we carry out a comprehensive study of the $J\ensuremath{-}K\ensuremath{-}\mathrm{\ensuremath{\Gamma}}$ model on a triangular lattice covering the whole parameter region, using a combination of exact diagonalization, classical Monte Carlo, and analytic methods, with an emphasis on the effects of the $\mathrm{\ensuremath{\Gamma}}$ term. In the HK limit ($\mathrm{\ensuremath{\Gamma}}=0$), we find five quantum phases which are quite similar to their classical counterparts. Among them, stripe-A and dual N\'eel phases are robust against the introduction of the $\mathrm{\ensuremath{\Gamma}}$ term, in particular, stripe A extends to the region connecting $K=\ensuremath{-}1$ and $K=1$ for $\mathrm{\ensuremath{\Gamma}}<0$. Though the ${120}^{\ensuremath{\circ}}$ N\'eel phase also extends to a finite $\mathrm{\ensuremath{\Gamma}}$, its region is largely reduced compared to the previous classical result. Interestingly, the ferromagnetic (dubbed FM-A) and stripe-B phases are unstable in response to an infinitesimal $\mathrm{\ensuremath{\Gamma}}$ interaction. Moreover, we find five additional phases for $\mathrm{\ensuremath{\Gamma}}\ensuremath{\ne}0$ which are elaborated by both the quantum and classical numerical methods. Part of the parameter space previously identified as the ${120}^{\ensuremath{\circ}}$ N\'eel phase in the classical study is found to give way to the modulated stripe phase. Depending on the sign of the $\mathrm{\ensuremath{\Gamma}}$ term, the FM-A phase transits into FM-B ($\mathrm{\ensuremath{\Gamma}}>0$) and FM-C ($\mathrm{\ensuremath{\Gamma}}<0$) phases with different spin orientations. Similarly, the stripe-B phase transits into stripe-C ($\mathrm{\ensuremath{\Gamma}}>0$) and stripe-A ($\mathrm{\ensuremath{\Gamma}}<0$) phases. Around the positive $\mathrm{\ensuremath{\Gamma}}$ point, due to the interplay of the Heisenberg, Kitaev, and $\mathrm{\ensuremath{\Gamma}}$ interactions, we find a possible quantum spin liquid in a noticeable region with a continuum in spin excitations.The celebrated Kitaev honeycomb model provides an analytically tractable example with an exact quantum spin liquid ground state. While in real materials, other types of interactions besides the Kitaev coupling ($K$) are present, such as Heisenberg ($J$) and symmetric off-diagonal ($\mathrm{\ensuremath{\Gamma}}$) terms, these interactions can also be generalized to a triangular lattice. Here, we carry out a comprehensive study of the $J\ensuremath{-}K\ensuremath{-}\mathrm{\ensuremath{\Gamma}}$ model on a triangular lattice covering the whole parameter region, using a combination of exact diagonalization, classical Monte Carlo, and analytic methods, with an emphasis on the effects of the $\mathrm{\ensuremath{\Gamma}}$ term. In the HK limit ($\mathrm{\ensuremath{\Gamma}}=0$), we find five quantum phases which are quite similar to their classical counterparts. Among them, stripe-A and dual N\'eel phases are robust against the introduction of the $\mathrm{\ensuremath{\Gamma}}$ term, in particular, stripe A extends to the region connecting $K=\ensuremath{-}1$ and $K=1$ for $\mathrm{\ensuremath{\Gamma}}<0$. Though the ${120}^{\ensuremath{\circ}}$ N\'eel phase also extends to a finite $\mathrm{\ensuremath{\Gamma}}$, its region is largely reduced compared to the previous classical result. Interestingly, the ferromagnetic (dubbed FM-A) and stripe-B phases are unstable in response to an infinitesimal $\mathrm{\ensuremath{\Gamma}}$ interaction. Moreover, we find five additional phases for $\mathrm{\ensuremath{\Gamma}}\ensuremath{\ne}0$ which are elaborated by both the quantum and classical numerical methods. Part of the parameter space previously identified as the ${120}^{\ensuremath{\circ}}$ N\'eel phase in the classical study is found to give way to the modulated stripe phase. Depending on the sign of the $\mathrm{\ensuremath{\Gamma}}$ term, the FM-A phase transits into FM-B ($\mathrm{\ensuremath{\Gamma}}>0$) and FM-C ($\mathrm{\ensuremath{\Gamma}}<0$) phases with different spin orientations. Similarly, the stripe-B phase transits into stripe-C ($\mathrm{\ensuremath{\Gamma}}>0$) and stripe-A ($\mathrm{\ensuremath{\Gamma}}<0$) phases. Around the positive $\mathrm{\ensuremath{\Gamma}}$ point, due to the interplay of the Heisenberg, Kitaev, and $\mathrm{\ensuremath{\Gamma}}$ interactions, we find a possible quantum spin liquid in a noticeable region with a continuum in spin excitations.

arXiv:2008.10972Physical review. B./Physical review. BFeb 4, 202122 citations

S

, we find three different symmetric

{Z}_{2}

spin liquid phases, separated by two continuous quantum phase transitions. Interestingly, the gap of elementary excitations remains finite throughout the transitions. The first spin liquid phase corresponds to the well-known zero-flux state in the Heisenberg limit, which is stable with respect to small Kitaev couplings and develops ${120}^{\ensuremath{\circ}}$ order in the semiclassical limit at large

S

. In the opposite Kitaev limit, we find a different spin liquid ground state, which is a quantum disordered version of a magnetically ordered state with antiferromagnetic chains, in accordance with results in the classical limit. Finally, at intermediate couplings, we find a spin liquid state with unusual spin correlations. Upon spinon condensation, this state develops Bragg peaks at incommensurate momenta in close analogy to the magnetically ordered Z2 vortex crystal phase, which has been analyzed in recent theoretical works.

Motivated by the possibility of finding a bosonic quantum spin liquid in the integer spin-

S

Heisenberg-Kitaev model on the honeycomb lattice, we derive a Schwinger boson mean-field theory involving both singlet and triplet pairing channels which includes hopping and pairing operators on equal footing. The mixed construction introduced here is justified by the good comparison with exact diagonalization energies of the $S\ensuremath{\le}3/2$ Heisenberg-Kitaev model and the perfect match with the Luttinger-Tisza semiclassical energies obtained at large

S

. We find various competing gapped quantum spin liquids close to the Kitaev point. A comparison of their spin excitation spectrum with the dynamical structure factor obtained from exact diagonalizations allows us to identify the physical spin liquid Ansatz of the model. In particular, we identify a chiral quantum spin liquid state whose spin excitation spectrum follows closely the exact diagonalization data and survives up to large spin $S\ensuremath{\lesssim}2$. We propose this state as a promising quantum spin liquid candidate for the integer spin-

S

antiferromagnetic Kitaev model which may be realized in

S=1

Kitaev materials

{\mathrm{A}}_{3}{\mathrm{Ni}}_{2}{\mathrm{XO}}_{6}

and

{\mathrm{KNiAsO}}_{4}

K

) are present, such as Heisenberg (

J

) and symmetric off-diagonal ($\mathrm{\ensuremath{\Gamma}}$) terms, these interactions can also be generalized to a triangular lattice. Here, we carry out a comprehensive study of the $J\ensuremath{-}K\ensuremath{-}\mathrm{\ensuremath{\Gamma}}$ model on a triangular lattice covering the whole parameter region, using a combination of exact diagonalization, classical Monte Carlo, and analytic methods, with an emphasis on the effects of the $\mathrm{\ensuremath{\Gamma}}$ term. In the HK limit ($\mathrm{\ensuremath{\Gamma}}=0$), we find five quantum phases which are quite similar to their classical counterparts. Among them, stripe-A and dual N\'eel phases are robust against the introduction of the $\mathrm{\ensuremath{\Gamma}}$ term, in particular, stripe A extends to the region connecting $K=\ensuremath{-}1$ and

K=1

for $\mathrm{\ensuremath{\Gamma}}<0$. Though the ${120}^{\ensuremath{\circ}}$ N\'eel phase also extends to a finite $\mathrm{\ensuremath{\Gamma}}$, its region is largely reduced compared to the previous classical result. Interestingly, the ferromagnetic (dubbed FM-A) and stripe-B phases are unstable in response to an infinitesimal $\mathrm{\ensuremath{\Gamma}}$ interaction. Moreover, we find five additional phases for $\mathrm{\ensuremath{\Gamma}}\ensuremath{\ne}0$ which are elaborated by both the quantum and classical numerical methods. Part of the parameter space previously identified as the ${120}^{\ensuremath{\circ}}$ N\'eel phase in the classical study is found to give way to the modulated stripe phase. Depending on the sign of the $\mathrm{\ensuremath{\Gamma}}$ term, the FM-A phase transits into FM-B ($\mathrm{\ensuremath{\Gamma}}>0$) and FM-C ($\mathrm{\ensuremath{\Gamma}}<0$) phases with different spin orientations. Similarly, the stripe-B phase transits into stripe-C ($\mathrm{\ensuremath{\Gamma}}>0$) and stripe-A ($\mathrm{\ensuremath{\Gamma}}<0$) phases. Around the positive $\mathrm{\ensuremath{\Gamma}}$ point, due to the interplay of the Heisenberg, Kitaev, and $\mathrm{\ensuremath{\Gamma}}$ interactions, we find a possible quantum spin liquid in a noticeable region with a continuum in spin excitations.