Charge Density Waves and the Hidden Nesting of Purple Bronze ${\mathrm{K}}_{0.9}{\mathrm{Mo}}_6{\mathrm{O}}_{17}$

We introduce the first multiorbital effective tight-binding model to describe the effect of electron-electron interactions in this system. Upon fixing all the effective hopping parameters in the normal state against an ab initio band structure, and with only the overall scale of the interactions as the sole adjustable parameter, we find that a self-consistent Hartree-Fock solution reproduces extremely well the experimental behavior of the charge density wave (CDW) order parameter in the full range $0<T<T_c$, as well as the precise reciprocal space locations of the partial gap opening and Fermi arc development. The interaction strengths extracted from fitting to the experimental CDW gap are consistent with those derived from an independent Stoner-type analysis.

Figure 1

(a) Top view of the two inner KMO sublayers of composition ${\mathrm{Mo}}_2{\mathrm{O}}_9$, and the full unit cell (inset); Mo, green and blue; O, red; K, purple. (b) Simplified representation of the slab shown in (a) with the oxygens removed and highlighting the effective 1D zigzag chains generated by the three degenerate Mo $d$ orbitals. (c) Diagram of the site, orbital, and hopping labels used in our tight-binding model.

Figure 2

(a) Calculated Fermi contours in the normal state ($T>T_c$, green lines) plotted together with the FS measured in Ref. [4]. ${\mathit{Q}}_{\mu }$ represent the three experimentally measured CDW wave vectors. (b) Intrasublattice and intraorbital susceptibility (${\chi }_d$) along $\mathrm{\Gamma }M$ at $T=115$ and 220 K. The dashed lines show ${\chi }_d$ calculated for decoupled 1D zigzag chains, while solid ones (vertically shifted by 0.1 for clarity) correspond to the full 2D TB model (see Supplemental Material for details [14]). (c) Overlaid ab initio (solid lines) and TB (dashed lines) bands.

Figure 3

Gap along $\mathrm{\Gamma }{K}^{\prime }$ as a function of the temperature. (a) Simplified treatment of the interactions (intraorbital interactions, but complete 2D hoppings) discussed in the text. $V_1=740\mathrm{meV}$ and $V_1=810\mathrm{meV}$ yield $T_c=115\mathrm{K}$ and $T_c=220\mathrm{K}$, respectively. (b) The two gaps along $\mathrm{\Gamma }{K}^{\prime }$ for the more realistic interacting Hamiltonian [see also Fig. 4]. The experimental data (points) are from Mou et al. [4].

Figure 4

(a) Self-consistent band structure in the CDW phase along the high symmetry directions of the folded zone at $T=0$ ($V_1=360\mathrm{meV}$, $V_2=-250\mathrm{meV}$, $\eta =0.2$). The rightmost panels amplify the dashed rectangles around $M^{\prime }$ and $\mathrm{\Gamma }{K}^{\prime }$, respectively. (b) Spectral function at $\omega =\mu$ (artificially broadened by 5 meV, folded zone). Ungapped portions of the FS (Fermi arcs) lie around $M^{\prime }$ (red, darker regions). (c) Noninteracting (main panel) and renormalized (inset) DOS.