Spin-triplet superconductivity in a weak-coupling Hubbard model for the quasi-one-dimensional compound ${\mathrm{Li}}_{0.9}{\mathrm{Mo}}_6{\mathrm{O}}_{17}$

The purple bronze ${\mathrm{Li}}_{0.9}{\mathrm{Mo}}_6{\mathrm{O}}_{17}$ is of interest due to its quasi-one-dimensional electronic structure and the possible Luttinger liquid behavior resulting from it. For sufficiently low temperatures, it is a superconductor with a pairing symmetry that is still to be determined. To shed light on this issue, we analyze a minimal Hubbard model for this material involving four molybdenum orbitals per unit cell near quarter filling, using asymptotically exact perturbative renormalization group methods. We find that spin-triplet odd-parity superconductivity is the dominant instability. Approximate nesting properties of the two quasi-one-dimensional Fermi surfaces enhance certain second-order processes, which play crucial roles in determining the structure of the pairing gap. Notably, we find that the gap has more sign changes than required by the point-group symmetry.

Figure 1

Tight-binding lattice model. Each circle corresponds to a single Mo atom and there are four atoms per unit cell. Filled and empty circles denote two types of crystallographically inequivalent Mo atoms, i.e., Mo(1) and Mo(4), within a layer of ${\mathrm{Li}}_{0.9}{\mathrm{Mo}}_6{\mathrm{O}}_{17}$. The horizontal orange line defines a plane of mirror symmetry, and the green dot an axis of $C_2$-rotation symmetry. (The reflection is only an approximate symmetry in the real material, which is slightly monoclinic [21].)

Figure 2

The Fermi surface and the sign structure of the (triplet) gap function for ${\eta }_w=1$. The inner and outer curves correspond to Fermi surface sheets associated with distinct bands. The vertical arrow depicts the nesting vector $\mathbf{Q}\equiv (0,0.95\pi )$. The oblique arrows denote an example of two points on the Fermi surface whose crystal momenta approximately sum to $\mathbf{Q}$; the gap function changes sign between these points although not required by symmetry.

Figure 3

Dominance of triplet pairing is insensitive to the extent of warping of the Fermi surface. The pairing strength in the triplet and singlet channels is shown as a function of the parameter ${\eta }_w$ that determines warping of the Fermi surface. Reference [11] estimates that ${\eta }_w\simeq 1$.

Figure 4

The pair wave function for ${\eta }_w=1$ on the inner and outer Fermi surfaces in the upper half of the Brillouin zone.