  We study the Kitaev-Heisenberg model on the triangular lattice by using the
two-dimensional density-matrix renormalization group method. Calculating the
ground-state energy and spin structure factors, we obtain the ground state
phase diagram of the Kitaev-Heisenberg model. As suggested by previous studies,
we find a 120$^\circ$ antiferromagnetic (AFM) phase, a $\mathbb{Z}_2$-vortex
crystal phase, a nematic phase, a dual $\mathbb{Z}_2$-vortex crystal phase
(dual counterpart of $\mathbb{Z}_2$-vortex crystal phase), a $\mathbb{Z}_6$
ferromagnetic phase, and a dual ferromagnetic phase (dual counterpart of
$\mathbb{Z}_6 $ ferromagnetic phase). Spin correlations discontinuously change
at the phase boundaries because of first-order phase transitions. In the (dual)
$\mathbb{Z}_6$ ferromagnetic phase, we find that the dominant spin component of
the spin structure factors is different on either side of the SU(2) symmetric
point, although there is no phase transition. Furthermore, we study the
relation among von Neumann entanglement entropy, entanglement spectrum, and
phase transitions of the model. We find that the Schmidt gap closes at the
phase boundaries and thus the entanglement entropy clearly changes as well.
This is different from the Kitaev-Heisenberg model on the honeycomb lattice,
where the Schmidt gap and the entanglement entropy are not necessary a good
measure of phase transitions.
