Instability of three-band Luttinger liquids: renormalization group analysis and possible application to K$_{\mathrm{2}}$Cr$_{\mathrm{3}}$As$_{\mathrm{3}}$

Motivated by recently discovered quasi-one-dimensional superconductor K$_{\mathrm{2}}$Cr$_{\mathrm{3}}$As$_{\mathrm{3}}$ with $D_{3h}_{\mathrm{\thinspace }}$lattice symmetry, we study one-dimensional three-orbital Hubbard models with generic electron repulsive interaction described by intra-orbital repulsion $U$, inter-orbital repulsion$U'$, and Hund's coupling $J$. As extracted from density functional theory calculation, two of the three atomic orbitals are degenerate and the third one is non-degenerate, and the system is presumed to be at incommensurate filling. With the help of bosonization, we have usual three-band Luttinger liquids in the normal state. Possible charge density wave (CDW), spin density wave (SDW) and superconducting instabilities are analyzed by one-loop renormalization group. The ground state depends on the ratio $J/U$. For the physical relevant parameter region, $\mbox{0}1/2$.