Charge fluctuations in the unconventional metallic state of ${\mathrm{Li}}_{0.9}{\mathrm{Mo}}_6{\mathrm{O}}_{17}$

Charge fluctuations in the quasi-one-dimensional material ${\mathrm{Li}}_{0.9}{\mathrm{Mo}}_6{\mathrm{O}}_{17}$ are analyzed based on a multiorbital extended Hubbard model. A charge-ordering transition induced by Coulomb repulsion is found with a particular charge-ordering pattern. The metallic phase is characterized by the presence of a charge collective mode which softens signaling the proximity to the charge order transition. We argue that the scattering between electrons mediated by these charge fluctuations can lead to non-Fermi liquid behavior in a quasi-one-dimensional system consistent with the unconventional metallic behavior observed above the superconducting transition in ${\mathrm{Li}}_{0.9}{\mathrm{Mo}}_6{\mathrm{O}}_{17}$.

Figure 1

Charge-ordering phenomena in the extended Hubbard model (1) for ${\mathrm{Li}}_{0.9}{\mathrm{Mo}}_6{\mathrm{O}}_{17}$. We show the crystal structure of ${\mathrm{Li}}_{0.9}{\mathrm{Mo}}_6{\mathrm{O}}_{17}$ projected onto the $b-c$ plane showing only the Mo and O atoms forming the zigzag ladders relevant to the low-energy electronic properties. The real-space charge-ordering pattern consisting of alternating charge-rich (purple) and charge-poor (magenta) Mo atoms arising in the U-V model is also shown.

Figure 2

Charge order instability induced by Coulomb repulsion in the U-V model. (a) The Fermi surface obtained from our effective model for ${\mathrm{Li}}_{0.9}{\mathrm{Mo}}_6{\mathrm{O}}_{17}$ is shown. (b) The static charge susceptibility ${\chi }_c\left(\mathbf{q}\right)$ along the $(0,\frac{q}{b},\frac{\pi }{c/2})$ direction shows the rapid increase of charge fluctuations at wave vector $\mathbf{Q}=(0,\frac{\pi }{b/2},\frac{\pi }{c/2})$ associated with the Coulomb-induced CO occurring at $V_{\mathrm{CO}}\approx 0.8$ eV. The smaller structure at $\mathbf{q}=(0,\frac{\pi }{b},\frac{\pi }{c/2})$ is related to Fermi surface nesting at $q=2k_F\lesssim \pi /b$. All energies are given in eV.

Figure 3

Phase diagram of the effective extended Hubbard U-V model for ${\mathrm{Li}}_{0.9}{\mathrm{Mo}}_6{\mathrm{O}}_{17}$. The T-V phase diagram obtained from RPA is shown for fixed value of $U=1$ eV and varying $V$. Charge-ordered (CO) and homogeneous metallic phases are separated by the CO transition line, $T_{\mathrm{CO}}$, which displays “reentrant” behavior. $T_m$ is the temperature scale associated with the onset of charge fluctuations. Fermi liquid (FL) and non-Fermi-liquid (NFL) phases are separated by the crossover scale $T^{*}$. The inset shows the $T$ dependence of $\mathrm{Im}{\chi }_c(\mathbf{Q},\omega )$ in the metallic phase close to CO displaying the softening and enhancement of the charge collective mode around $T_m$. All energies are given in eV.

Figure 4

Phase diagram of an extended Hubbard model including up to the third-neighbor Coulomb interactions relevant for ${\mathrm{Li}}_{0.9}{\mathrm{Mo}}_6{\mathrm{O}}_{17}$. The T-V phase diagram obtained from RPA is shown for fixed value of $U=2$ eV and varying $V$ for the model including up to third-nearest-neighbor Coulomb interactions. The phase diagram shows similar qualitative behavior to that in Fig. 3 with suppressed energy scales due to CO frustration effects. We also show here the low-temperature $2k_F$-type of instability occurring at low temperatures. The critical value for the CO driven by Coulomb repulsion is $V_c=0.66$ as $T\to 0$. The suggested location for ${\mathrm{Li}}_{0.9}{\mathrm{Mo}}_6{\mathrm{O}}_{17}$ at ambient pressure is marked with a vertical arrow. The system would effectively shift away from the CO transition under pressure. All energies are given in eV.

Figure 5

Imaginary part of the charge susceptibility $\mathrm{Im}{\chi }_c(q,\omega )$ of the U-V model showing the emergence and softening of the collective excitation as the interaction increases. (a) Noninteracting charge susceptibility $\mathrm{Im}{\chi }_0(q,\omega )$ displaying the particle-hole continuum, (b) a Hubbard-like interaction ($U=1$ eV, $V=0$), (c) an interaction compatible with purple bronze phenomenology $U=1$ eV, $V=0.68$ eV, and (d) close to the CO transition $U=1,V=0.79$ eV. Temperature is $T=0.05$ eV. All energies are given in eV.

Figure 6

Crossover temperature $T_{\mathrm{LL}}$ and exponent of the density of states $\alpha$ estimated as a function of the coupling $V$.