Unconventional orbital charge density wave mechanism in the transition metal dichalcogenide $1T-{\mathrm{TaS}}_2$

The transition metal dichalcogenide $1T-{\mathrm{TaS}}_2$ is attracting growing attention because of the formation of rich density wave (DW) and superconducting transitions. However, the origin of the incommensurate DW state at the highest temperature ($\sim 550\mathrm{K}$), which is “the parent state” of the rich physical phenomena, is still uncovered. Here, we present a natural explanation for the triple-$\mathit{q}$ incommensurate DW in $1T-{\mathrm{TaS}}_2$ based on the first-principles Hubbard model with on-site $U$. We apply the paramagnon interference mechanism that gives the nematic order in Fe-based superconductors. The derived order parameter has very unique characters: (i) an orbital-selective nature, and (ii) an unconventional sign reversal in both momentum and energy spaces. The present Letter will be useful for understanding the rich physics in $1T-{\mathrm{TaS}}_2$, $1T-{\mathrm{VSe}}_2$, and other transition metal dichalcogenides.

Figure 1

(a) Three electron-type FSs and the intra-FS nesting vectors ${\mathit{q}}_n$ ($n=1,2,3$). Here, ${\mathit{q}}_1=(0.56\pi ,0)$. (b) Paramagnon interference mechanism of charge/orbital order at wave vector ${\mathit{q}}_1$. (c) Momentum conservation law in the interference mechanism.

Figure 2

(a) Diagrammatic expression of the DW equation. (b) Obtained eigenvalue of the DW equation ${\lambda }_{\mathit{q}}$ for $T=40\mathrm{meV}$ and ${\alpha }_S=0.85$. ${\lambda }_{\mathit{q}}$ shows the peaks at the intra-FS nesting vectors $q={\mathit{q}}_n$ ($n=1,2,3$), consistently with experiments. Here, ${\mathit{q}}_1=(0.56\pi ,0)$. (c) Eigenvalue ${\lambda }_{{\mathit{q}}_n}$ as a function of ${\alpha }_S$. It reaches unity for ${\alpha }_S\gtrsim 0.90$.

Figure 3

(a) Real part of obtained form factor $f_{{\mathit{q}}_1}^{1,2}\left(\mathit{k}\right)$, which is the most significant component for the IC-CDW at $\mathit{q}={\mathit{q}}_1$. (b) Fourier transformation of the form factor $f_{{\mathit{q}}_1}^{1,2}\left(\mathit{r}\right)$. The signal at $\mathit{r}=\mathit{0}$ represents the local $\mathrm{orbital}+\mathrm{charge}$ order with respect to orbitals 1 and 2, and the signal at $\mathit{r}\ne \mathit{0}$ exhibits the nonlocal bond order. The strongest bond order is at $\mathit{r}=(0,\pm 2)$ marked by pink dashed circles.

Figure 4

(a) $\mathrm{Re}{f}_{{\mathit{q}}_1}^{1,2}(\mathit{k},{\epsilon }_n)$ as a function of ${\epsilon }_n$ at $\mathit{k}=(0,0.5\pi )$. (b) Diagrammatic expression of the fourth-order term in the GL equation. (c) Triple-$\mathit{q}$ IC-CDW state in real space.

Figure 5

(c) Unfolded FS in the single-${\mathit{q}}_1$ state and (d) that in the triple-$\mathit{q}$ state. (c) Reason for the FS recovery in the triple-$\mathit{q}$ CDW, which occurs when the state $|\mathit{k}\rangle$ hybridizes with $|\mathit{k}+{\mathit{q}}_1\rangle$ and $|\mathit{k}-{\mathit{q}}_2\rangle$ simultaneously. (d) DOS as a function of $\epsilon -E_{\mathrm{F}}$. We introduced a BCS-type cutoff ${\omega }_c=4f^{\max }$.