Triplet superconductivity in a model of ${\text{Li}}_{0.9}{\text{Mo}}_6{\text{O}}_{17}$

Superconductivity in the quasi-one-dimensional material ${\mathrm{Li}}_{0.9}{\mathrm{Mo}}_6{\mathrm{O}}_{17}$ is analyzed based on a multiorbital extended Hubbard model. We found strong charge fluctuations at two different momenta ${\mathbf{Q}}_{\mathbf{1}}$ and ${\mathbf{Q}}_{\mathbf{2}}$ giving rise to two different charge-ordered phases. Evaluating the superconducting vertex, we found superconductivity near strong charge fluctuations at ${\mathbf{Q}}_{\mathbf{1}}$. The order parameter has $p$-wave symmetry with nodes on the Fermi surface. The metallic state displays a characteristic charge collective mode ${\mathbf{Q}}_{\mathbf{1}}$ due to nesting and, for on-site Hubbard repulsion sufficiently large, a charge critical mode ${\mathbf{Q}}_{\mathbf{2}}$ driven by Coulomb repulsion, which softens at the proximity to the transition. The results are quite robust for different coupling parametrizations. A phase diagram discussing the relevance of the model to the physics of the material is proposed.

Figure 1

Schematic crystal structure of ${\mathrm{Li}}_{0.9}{\mathrm{Mo}}_6{\mathrm{O}}_{17}$ projected onto the $b\text{−}c$ plane showing only the partially filled Mo atoms forming the zigzag ladders relevant to the low-energy electronic properties. Our choice of unit cell is highlighted and the orbitals numerated (solid line) according to the text; the hoppings (dotted line) and the Coulomb interactions (dashed line) are also represented.

Figure 2

(a) Fermi surface with two bands, ${\mathbf{Q}}_{\mathbf{1}}$ a nesting vector, and ${\mathbf{Q}}_{\mathbf{2}}$ referred to in the text. (b) Real part of the bare susceptibility in momentum space for $\omega =0$ and $q_a=0$. Notice the maximum reveals the warping of the Fermi surface at the nesting vector.

Figure 3

(a) Phase diagram $U\text{−}V$. Extended region of $p_y$ superconductivity with nodes in the Fermi surface (inset) close to the CO region. In the inset we show $d_{x^2-y^2}$ and $p_y$ wave functions.

Figure 4

Left: Momentum space distribution of the interaction, showing the sum of all components. Right: The same as in Fig. 5 (bottom right), in those momenta relevant to the pairing vertex.

Figure 5

Top: Imaginary part of the larger eigenvalue of the charge susceptibility near the critical value (see Fig. 3) for (left) $U=0.2$ and (right) $U=0.4$. Bottom: Real part of the larger eigenvalue in momentum space and zero frequency close to the critical value for (left) $U=0.2$ and (right) $U=0.4$.

Figure 6

Gap squared of the ${\mathbf{Q}}_{\mathbf{2}}$ critical mode scaled with the critical interaction $V=V_c$. We observe a $\frac{1}{2}$ exponent near the critical value.

Figure 7

The background represents the charge fluctuations near ${\mathbf{Q}}_{\mathbf{1}}$ and on top of that are the CO transition line (green) and superconducting transition line (solid black) for (left) $U=0.2$ and (right) $U=0.4$. The dashed black line is the CO transition due to ${\mathbf{Q}}_{\mathbf{1}}$ if the other order is not present. The blue line is the superconducting transition.

Figure 8

Left: Gap in the Matsubara frequency $\left(i{\omega }_n\right)$ at $T=0.01,U=0.2$, and $V=0.676$, green line; Lorentzian fit, blue line. Right: Conductance against energy for different superconducting order parameters. We can distinguish the $p_y$ wave; $\alpha$ is the angle of the order parameter with the junction, and $Z$ is the height of the tunnel barrier, in this case the insulating phase [40].

Figure 9

Schematic phase diagram for LiPB. The present study comprises the green horizontal arrow; we believe the temperature dependence of the real material is represented by the vertical orange arrow.

Figure 10

Dashed line, renormalization group estimation of the crossover temperature for the critical $V$ values found for each $U$; solid line, the exponent in the Luttinger density of states, $\alpha$.

Figure 11

Normalized imaginary part of the charge susceptibility at ${\mathbf{Q}}_{\mathbf{1}}$ weighted by ${\omega }^{-1}$ against frequency for different interaction parameters. $U=0.2$ (left) and $U=0.4$ (right) and $V$ at the superconducting transition and CDW or CO transition. Dashed lines show the energy at which bands not taken into account in the model would contribute at such internal momentum.