Spin-orbit coupling and odd-parity superconductivity in the quasi-one-dimensional compound ${\mathrm{Li}}_{0.9}{\mathrm{Mo}}_6{\mathrm{O}}_{17}$

Previous theoretical studies [W. Cho, C. Platt, R. H. McKenzie, and S. Raghu, Phys. Rev. B 92, 134514 (2015); N. Lera and J. V. Alvarez, Phys. Rev. B 92, 174523 (2015)] have suggested that ${\mathrm{Li}}_{0.9}{\mathrm{Mo}}_6{\mathrm{O}}_{17}$, a quasi-one-dimensional “purple bronze” compound, exhibits spin-triplet superconductivity and that the gap function changes sign across the two nearly degenerate Fermi surface sheets. We investigate the role of spin-orbit coupling (SOC) in determining the symmetry and orientation of the $d$ vector associated with the superconducting order parameter. We propose that the lack of local inversion symmetry within the four-atom unit cell leads to a spin-orbit coupling analogous to that proposed for graphene, ${\mathrm{MoS}}_2$, or SrPtAs. In addition, from a weak-coupling renormalization group treatment of an effective model Hamiltonian, we find that SOC favors the odd parity $A_{1u}$ state with $S_z=\pm 1$ over the $B$ states with $S_z=0$, where $z$ denotes the least-conducting direction. We discuss possible definitive experimental signatures of this superconducting state.

Figure 1

Tight-binding lattice model. Each circle corresponds to a single Mo atom and there are four atoms per unit cell (gray rectangle). Filled and empty circles denote two types of crystallographically inequivalent Mo atoms, i.e., Mo(1) and Mo(4), within the layers of ${\mathrm{Li}}_{0.9}{\mathrm{Mo}}_6{\mathrm{O}}_{17}$. Intrachain, intraladder, and interladder hopping integrals are denoted $t,t_{\perp }$, and $t^{\prime }$, respectively. SOC is represented by a spin-dependent hopping term $+i\lambda {\sigma }^z$ $\left(-i\lambda {\sigma }^z\right)$ along (opposite) the arrow directions within the chains. The two yellow circles define centers of $C_2$ rotational symmetry; the blue (red) lines indicate glide mirror (mirror) planes.

Figure 2

First and second order contribution to the effective interaction $V_{\text{eff}}$. Each solid line corresponds to a propagator with momentum $k$, band index $b$, and spin $s$. The dashed line refers to the interaction in (6) and provides nonzero contributions for spin configurations of the type indicated in diagram (1a).

Figure 3

Leading dimensionless eigenvalues ${\tilde{w}}_{0i}\equiv w_{0i}(W^2/U^2)$ for the even-parity, odd parity $B_{3u}$ $\left(S_z=0\right)$ and for the degenerate $A_{1u}/B_{2u}$ $\left(S_z=\pm 1\right)$ channel as a function of the spin-orbit coupling $\lambda$.

Figure 4

(left to right) Leading eigenvectors in the even parity $\left(S_z=0\right)$, odd parity $B_{3u}$ $\left(S_z=0\right)$, and in the degenerate $A_{1u}/B_{2u}$ $\left(S_z=\pm 1\right)$ channel. The upper row displays the case of zero spin-orbit coupling $\lambda =0.0t$, the lower one shows $\lambda =0.03t$. The upper (lower) subdivision in each viewgraph corresponds to the upper (lower) Fermi surface of equal color coding.