Spin-Fluctuation-Driven Nematic Charge-Density Wave in Cuprate Superconductors: Impact of Aslamazov-Larkin Vertex Corrections

We present a microscopic derivation of the nematic charge-density wave (CDW) formation in cuprate superconductors based on the three-orbital $d\text{−}p$ Hubbard model by introducing the vertex correction (VC) into the charge susceptibility. The CDW instability at $\mathit{q}=({\mathrm{\Delta }}_{\text{FS}},0)$, $(0,{\mathrm{\Delta }}_{\text{FS}})$ appears when the spin fluctuations are strong, due to the strong charge-spin interference represented by the VC. Here, ${\mathrm{\Delta }}_{\text{FS}}$ is the wave number between the neighboring hot spots. The obtained spin-fluctuation-driven CDW is expressed as the “intra-unit-cell orbital order” accompanied by the charge transfer between the neighboring atomic orbitals, which is actually observed by the scanning tunneling microscope measurements. We predict that the cuprate CDW and the nematic orbital order in Fe-based superconductors are closely related spin-fluctuation-driven phenomena.

Figure 1

(a) Three-orbital $d\text{−}p$ model for the ${\mathrm{CuO}}_2$ plane. (b) FS for $x=0.1$. The integrand in Eq. (3) is large when three points $\mathit{k}-\mathit{q}/2$, $\mathit{k}+\mathit{q}/2$, $\mathit{k}-\mathit{p}$ ($\mathit{p}={\mathit{Q}}_s$) are connected by the nesting vectors. (c) ${\chi }_d^s(\mathit{q})$ given by the RPA. The unit is ${\mathrm{eV}}^{-1}$. We put $U_d=4.06\mathrm{eV}$ and $U_p=0$. (d) Diagrammatic expression of $X_{l;m}^c(\mathit{q})$. (e) AL-VCs $X_x^c(\mathit{q})$ and $X_y^c(\mathit{q})$. (f) $C^s(\mathit{q})$ and $|{\mathrm{\Lambda }}_3(\mathit{q};{\mathit{Q}}_s){|}^2$ as functions of $\mathit{q}$.

Figure 2

Charge susceptibilities with the VC: (a) ${\chi }_d^c(\mathit{q})$, (b) ${\chi }_x^c(\mathit{q})$ and ${\chi }_y^c(\mathit{q})$, (c) ${\chi }_{d;x}^c(\mathit{q})$ and ${\chi }_{d;y}^c(\mathit{q})$. We put $U_d=4.06\mathrm{eV}$, $U_p=0$, and $V=0.65\mathrm{eV}$. (d) A possible charge pattern of the CDW (${\delta }_c=\pi /2$). Since the charge transfer between the neighboring $n_y$ and $n_d$ occurs, ($n_y$, $n_d$) are in antiphase in the intra-unit cell. (e) Total DOS and local DOSs at two $p_y$ sites in the CDW state with ${\mathit{Q}}_c=(\pi /2,0)$. (f) $R(\mathit{r},E)$ in the CDW state for $E=0.2\mathrm{eV}$.

Figure 3

(a) ${\mathrm{\Delta }}_{\text{FS}}$, ${\delta }_s$, and ${\delta }_c$ obtained in the $d\text{−}p$ model as functions of $x=5-n$. (Inset) The values of $V$ required to give the CDW. (b) The values of $V_{\mathrm{CDW}}^{-1}$ as a function of $x$, which will depict a qualitative behavior of $T_{\mathrm{CDW}}$. Note that ${\mathit{Q}}_c$ shifts to $\mathbf{0}$ for $x\le 0.02$. (Inset) ${\alpha }_S(x)$ (i)–(iii) are shown. (c) Schematic phase diagram of cuprates.

Figure 4

(a) $C^s(\mathit{q})$ for $x=0.1$ and (b) ${\delta }_s$ and ${\delta }_c$ in the case of ${\alpha }_S=0.998$ ($U_d=4.09\mathrm{eV}$) and $V_{\mathrm{CDW}}\approx 0.35\mathrm{eV}$.