  The $120^{\circ}$ antiferromagnetism (AFM) is widely believed as the magnetic
ground state of the triangular systems because of the geometrical frustration.
The emergence of novel magnetism, such as the row-wise AFM in Mn/Cu(111) and
Sn/Si(111), reveals the importance of the longer-range hopping on magnetic
competitions in realistic material systems. By utilizing advanced many-body
techniques, we systematically studied the isotropic triangular Hubbard model
with second nearest-neighbor hopping $t^{\prime}$, including both the single-
and the two-particle responses. We found that both electronic and magnetic
phase transitions show a clear dependence on $t^{\prime}/t$. Consequently, we
observed a remarkable reentrance of the metal-insulator transition and a
crossover between the $120^{\circ}$- and the row-wise AFM. The Fermi surface
(FS) shows two distinct structures with the nesting vectors consistent with the
magnetic correlations. When $t^{\prime}$ evolves from 0 to 1, the correlated
Fermi surface demonstrates a Lifschitz transition between the two nesting
structures, and exotic phases like the featureless insulating state can be
realized. Our work sheds light on the engineering of electronic and magnetic
correlations of correlated triangular surfaces via longer-range hopping. The
rich phase diagram and the high degree of tunability make the triangular
lattice with longer-range hopping a more realistic platform to study the
emergent magnetic competitions.
