Thermal Properties and Instability of a U(1) Spin Liquid on the Triangular Lattice

We study the effect of Dzyaloshinskii-Moriya (DM) interaction on the triangular lattice U(1) quantum spin liquid (QSL) which is stabilized by ring-exchange interactions. A weak DM interaction introduces a staggered flux to the U(1) QSL state and changes the density of states at the spinon Fermi surface. If the DM vector contains in-plane components, then the spinons gain nonzero Berry phase. The resultant thermal conductances ${\kappa }_{xx}$ and ${\kappa }_{xy}$ qualitatively agree with the experimental results on the material ${\mathrm{EtMe}}_3\mathrm{Sb}[\mathrm{Pd}(\mathrm{dmit}{)}_2{]}_2$. Furthermore, owing to perfect nesting of the Fermi surface, a spin density wave state is triggered by larger DM interactions. On the other hand, when the ring-exchange interaction decreases, another antiferromagnetic (AFM) phase with 120° order shows up which is proximate to a U(1) Dirac QSL. We discuss the difference of the two AFM phases from their static structure factors and excitation spectra.

Figure 1

Phase diagram of the model (1), which includes two magnetically ordered phases and one U(1) QSL phase. The U(1) QSL has a finite spinon Fermi surface, while the AFM-I and AFM-II phase exhibit 120° order and approximate 120° order respectively, and are separated by a sharp first order phase transition. The inset illustrates the pattern of the DM interaction, where the vector ${\widehat{\mathit{D}}}_{ij}$ is along the $\widehat{\mathit{z}}/-\widehat{\mathit{z}}$ direction if the bond $i\to j$ is along or against the arrows.

Figure 2

(a) DOS at the Fermi surface as a function of magnetic field $B_z$. (b) Mean field gauge charge-charge and charge-spin Hall conductances ${\sigma }_{xy}^{cc}$ and ${\sigma }_{xy}^{cs}$ as functions of $B_z$ (here we assume that the angle between ${\widehat{\mathit{D}}}_{ij}$ and $\widehat{\mathit{z}}$ is $\pi /16$). (c) Cartoon picture of the thermal Hall conductance, where $e_x$ and $e_y$ are the emergent gauge ‘electric’ field. (d) The calculated thermal Hall angle as a function of $B_z$.

Figure 3

(a) and (b) The unit cells of the AFM-I and AFM-II phases, respectively. In AFM-I, the singlet hopping terms contain $0/\pi$ flux on the white or dark triangles, and the triplet hopping terms contain uniform $\pi /2$ flux. In AFM-II, the singlet or triplet hopping terms see staggered ${\varphi }_s$ flux or ${\varphi }_t$ flux. (c) and (d), The one-magnon bands and continuum spectra (in a system with 72 sites) for AFM-I ($D/J=0,J_r/J=0$) and AFM-II ($D/J=0.1,J_r/J=0.35$), respectively. The dashed line is the energy of the ground state before the renormalization, while the red dot stands for the renormalized one. The inset of (c) shows the dynamic structure factor of the Heisenberg model, and the inset of (d) illustrates the Brillouin Zone and the path of the dispersion curves.