Instability of three-band Tomonaga-Luttinger liquid: renormalization group analysis and possible application to K2Cr3As3

TL;DR

This paper analyzes the stability and possible phases of a three-band Tomonaga-Luttinger liquid model relevant to the superconductor K$_{2}$Cr$_{3}$As$_{3}$, revealing how interactions influence superconductivity and density wave states.

Contribution

It provides a renormalization group analysis of a three-orbital Hubbard model with degeneracy considerations, identifying conditions for various superconducting and density wave instabilities.

Findings

Spin-singlet SC favored at J<U/3

Spin-triplet SC favored at U/3<J<U/2

SDW instability dominates when degeneracy is lifted

Abstract

Motivated by recently discovered quasi-one-dimensional superconductor K$_{2}$Cr$_{3}$As$_{3}$ with $D_{3h}$ lattice symmetry, we study one-dimensional three-orbital Hubbard model with generic electron repulsive interaction described by intra-orbital repulsion $U$, inter-orbital repulsion, 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 non-degenerate ($A^{\prime}_1$), and the system is presumed to be at an incommensurate filling. With the help of bosonization, we have usual three-band Tomonaga-Luttinger liquid for the normal state. Possible charge density wave (CDW), spin density wave (SDW) and superconducting (SC) instabilities are analyzed by renormalization group. The ground state depends on the ratio $J/U$ and is sensitive to the degeneracy of $E^{\prime}$ bands. At $0<J<U/3$, spin-singlet SC state is favored, while spin-triplet superconductivity will be 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 two-fold degeneracy of $E^{\prime}$ bands is lifted, 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 $2\Delta k_F x$ at $U/3<J<U/2$. Possible experimental consequences and applications to K$_{2}$Cr$_{3}$As$_{3}$ are discussed.