Competing states in the $S=1/2$ $J_1$-$J_2$ Heisenberg model on the triangular lattice: a dynamical density-matrix renormalization group study

We improve and utilize the dynamical density-matrix renormalization group method to investigate the intermediate $J_2$ region in the $S=1/2$ triangular-lattice $J_1$-$J_2$ Heisenberg model, which has been argued to host either a gapped $\mathbb{Z}_2$ quantum spin liquid (QSL) or a gapless Dirac QSL. We find compelling evidence for the existence of two distinct competing states with markedly different low-energy excitations, obtained with different initial states across cylinders with circumferences ranging from 6 to 9. The state with slightly higher energy possesses dynamical features consistent with a Dirac QSL. For the lower-energy state, both the static and dynamical correlations are qualitatively similar to the magnetically ordered state at $J_2=0$, indicating it is probably either a weakly magnetically ordered state or a gapped $\mathbb{Z}_2$ QSL that is continuously connected to the ordered state.