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We study the ground state properties of the Hubbard model on three-legtriangular cylinders using large-scale density-matrix renormalization groupsimulations. At half-filling, we identify an intermediate gapless spin liquidphase between a metallic phase at weak coupling and Mott insulating dimer phaseat strong interaction, which has one gapless spin mode and algebraic spin-spincorrelations but exponential decay scalar chiral-chiral correlations. Uponlight doping the gapless spin liquid, the system exhibits power-lawcharge-density-wave (CDW) correlations but short-range single-particle,spin-spin, and chiral-chiral correlations. Similar to CDW correlations, thesuperconducting correlations are also quasi-long-ranged but oscillate in signas a function of distance, which is consistent with the striped pair-densitywave. When further doping the gapless spin liquid phase or doping the dimerorder phase, another phase takes over, which has similar CDW correlations butall other correlations decay exponentially.

Gapless spin liquid and pair density wave of the Hubbard model on three-leg triangular cylinders