Nematicity, magnetism, and superconductivity in FeSe under pressure: Unified explanation based on the self-consistent vertex correction theory

To understand the rich electronic phase diagram in FeSe under pressure that vividly demonstrates the strong interplay between the nematicity, magnetism, and superconductivity, we analyze the electronic states by including the higher-order many-body effects called the vertex correction (VC). We predict the pressure-induced emergence of $xy$-orbital hole pocket based on the first-principles analysis. Due to this pressure-induced Lifshitz transition, the spin fluctuations on the $xy$ orbital are enhanced, whereas those on $xz,yz$ orbitals are gradually reduced. For this reason, nonmagnetic orbital order $O=n_{xz}-n_{yz}$, which is driven by the spin fluctuations on $xz,yz$ orbitals through the intraorbital VCs, is suppressed, and it is replaced with the magnetism of $xy$-orbital $d$ electrons. The nodal $s$-wave state at ambient pressure ($O\ne 0$) and the enhancement of $T_{\mathrm{c}}$ under pressure are driven by the cooperation between the spin and orbital fluctuations.

Figure 1

(a) Band structure of the present FeSe model for $\mathrm{\Delta }{E}_{xy}=0$, 0.06, and 0.12 eV. (b) FSs for $\mathrm{\Delta }{E}_{xy}=0$ and (c) FSs for $\mathrm{\Delta }{E}_{xy}=0.12$ eV. The hFS3 is the $xy$-orbital hole pocket. The colors green, red, and blue represent the $xz, yz$, and $xy$ orbitals, respectively.

Figure 2

(a) Charge-channel AL-type VC for the $yz$-orbital and (b) three-point vertex. (c) ${\chi }_{xy}^s\left(\mathit{q}\right)$, (d) ${\chi }_{yz}^s\left(\mathit{q}\right)$, and (e) ${\chi }^{\mathrm{orb}}\left(\mathit{q}\right)$ or $\mathrm{\Delta }{E}_{xy}=0$ and 0.12 eV at $r=0.266$ and $T=30$ meV. (f) Stoner factors as functions of $\mathrm{\Delta }{E}_{xy}$ at $r=0.266$ and $T=30$ meV. (g) $T$ dependences of ${\alpha }_{S,C}$ for $\mathrm{\Delta }{E}_{xy}=0.08$ eV and (h) those for $\mathrm{\Delta }{E}_{xy}=0.12$ eV. (i) $T_m$ and $T_{\mathrm{str}}$ defined by the relation ${\alpha }_{S,C}=1$ and 0.96.

Figure 3

(a) $V_{\left(\mathrm{Migdal}\right)}$, (b) $V^{\mathrm{\Lambda }}$, and (c) $V^{\mathrm{cross}}$. (d) $U$-VC ${\mathrm{\Lambda }}^x$. (e) Eigenvalues for various pairing interactions: ${\lambda }^{\mathrm{tot}}$ for $V^{\mathrm{tot}}, {\lambda }^{\mathrm{\Lambda }}$ for $V^{\mathrm{\Lambda }}$, and ${\lambda }_{\left(\mathrm{Migdal}\right)}$ for $V_{\left(\mathrm{Migdal}\right)}$. (f) $s_{\pm }$-wave gap given by $V^{\mathrm{\Lambda }}$ for $\mathrm{\Delta }{E}_{xy}=0.12$ eV. (g) $s_{++}$-wave gap given by $V^{\mathrm{tot}}$. The FSs are shown in Fig. 1. Numerical study is performed in the nonmagnetic state.

Figure 4

(a) FSs and band structure in the presence of the sign-reversing orbital polarization $\delta E_{\mathrm{nem}}\left(\mathrm{\Gamma }\right)=-30$ meV and $\delta E_{\mathrm{nem}}\left(\mathrm{X}\right)=60$ meV. Here, $\mathrm{\Delta }{E}_{xy}=0$. (b) ${\chi }_{xz}^s\left(\mathit{q}\right), {\chi }_{yz}^s\left(\mathit{q}\right), {\chi }_{xz}^c\left(\mathit{q}\right)$, and ${\chi }_{yz}^c\left(\mathit{q}\right)$ obtained by the SC-VC theory. (c) $s$-wave gap functions obtained for $V^{\mathrm{tot}}$: the gap functions on each FS are nodal.