The Hubbard model and its strong-coupling version, the Heisenberg one, have been widely studied on the triangular lattice to capture the essential low-temperature properties of different materials. One example is given by transition metal dichalcogenides, as 1T-TaS2, where a large unit cell with 13 Ta atoms forms weakly coupled layers with an isotropic triangular lattice. By using accurate variational Monte Carlo calculations, we report the phase diagram of the t-t′ Hubbard model on the triangular lattice, highlighting the differences between positive and negative values of t′/t; this result can be captured only by including the charge fluctuations that are always present for a finite electron-electron repulsion. Two spin-liquid regions are detected: one for t′/t<0, which persists down to intermediate values of the electron-electron repulsion, and a narrower one for t′/t>0. The spin-liquid phase appears to be gapless, though the variational wave function has a nematic character, in contrast to the Heisenberg limit. We do not find any evidence for nonmagnetic Mott phases in the proximity of the metal-insulator transition, at variance with the predictions (mainly based upon strong-coupling expansions in t/U) that suggest the existence of a weak-Mott phase that intrudes between the metal and the magnetically ordered insulator.
