Motivated by recently discovered quasi-one-dimensional superconductor ${\mathrm{K}}_2{\mathrm{Cr}}_3{\mathrm{As}}_3$ with $D_{3h}$ lattice symmetry, we study a one-dimensional three-orbital Hubbard model with generic electron repulsive interaction described by intraorbital repulsion $U$, interorbital repulsion $U^{\prime }$, and Hund's coupling $J$. As extracted from density functional theory calculation, two of the three atomic orbitals are degenerate ($E^{\prime }$ states) and the third one is nondegenerate ($A_1^{\prime }$), and the system is at incommensurate filling. With the help of bosonization, the normal state is described by a three-band Tomonaga-Luttinger liquid. Possible charge density wave (CDW), spin density wave (SDW), and superconducting (SC) instabilities are analyzed by renormalization group method. The ground state depends on the ratio $J/U$ and is sensitive to the degeneracy of $E^{\prime }$ bands. The spin-singlet SC state is favored at $0<J<U/3$, and the spin-triplet SC state is favored in the region $U/3<J<U/2$. The SDW state has the lowest energy only in the unphysical parameter region $J>U/2$. When the twofold degeneracy of $E^{\prime }$ bands is lifted, the SDW instability has the tendency to dominate over the spin-singlet SC state at $0<J<U/3$, while the order parameter of the spin-triplet SC state will be modulated by a phase factor $2\mathrm{\Delta }{k}_{F}x$ at $U/3<J<U/2$. Possible experimental consequences and applications to ${\mathrm{K}}_2{\mathrm{Cr}}_3{\mathrm{As}}_3$ are discussed.

• Received 28 March 2016
• Revised 26 September 2016

DOI:https://doi.org/10.1103/PhysRevB.94.205129