Interatomic Coulomb interaction and electron nematic bond order in FeSe

Despite having the simplest atomic structure, bulk FeSe has an observed electronic structure with the largest deviation from the band theory predictions among all Fe-based superconductors and exhibits a low-temperature nematic electronic state without intervening magnetic order. We show that the Fe-Fe interatomic Coulomb repulsion $V$ offers a natural explanation for the puzzling electron correlation effects in FeSe superconductors. It produces a strongly renormalized low-energy band structure where the van Hove singularity sits remarkably close to Fermi level in the high-temperature electron liquid phase as observed experimentally. This proximity enables the quantum fluctuations in $V$ to induce a rotational symmetry breaking electronic bond order in the $d$-wave channel. We argue that this emergent low-temperature $d$-wave bond nematic state, different from the commonly discussed ferro-orbital order and spin nematicity, has been observed recently by several angle-resolved photoemission experiments detecting the lifting of the band degeneracies at high-symmetry points in the Brillouin zone. We present a symmetry analysis of the space group and identify the hidden antiunitary $T$ symmetry that protects the band degeneracy and the electronic order/interaction that can break the symmetry and lift the degeneracy. We show that the $d$-wave nematic bond order, together with the spin-orbit coupling, provide a unique explanation of the temperature dependence, momentum space anisotropy, and domain effects observed experimentally. We discuss the implications of our findings on the structural transition, the absence of magnetic order, and the intricate competition between nematicity and superconductivity in FeSe superconductors.

Figure 1

(a) LDA band structure of FeSe. (b) Band dispersions observed by ARPES in the symmetric phase at $120\mathrm{K}$. (c) Band degeneracy at high-symmetry points ($\mathrm{\Gamma },M$) among $d_{xz,e}$ (blue), $d_{yz,e}$ (red), $d_{xz,o}$ (cyan), and $d_{yz,o}$ (green). (d) ARPES results in the nematic state at $22\mathrm{K}$. Red dashed lines near $\mathrm{\Gamma }$ are data taken at $120\mathrm{K}$, showing ${\mathrm{\Delta }}_{\mathrm{\Gamma }}$ is nearly $T$-independent. Solid and dashed blue lines near $M$ correspond to the two domains.

Figure 2

(a) TB band of FeSe. Black solid lines are from $P\left(k\right)$ and red dashed lines are from $P(k+Q)$. (b) BZ of FeSe. The larger zone bounded by the black lines is the one-Fe BZ, while the smaller zone with blue boundaries is the two-Fe BZ. $Q$ corresponds to one of reciprocal lattice vectors in the two-Fe BZ. The dashed black lines with arrows indicate the trajectory over which the band dispersions are plotted in (a).

Figure 3

Renormalized band structure (a)–(d) in unit of eV and corresponding FS (e)–(h) at $V=0$ (a), $0.73\mathrm{eV}$ (b), $0.763\mathrm{eV}$ (c), and $0.85\mathrm{eV}$ (d). Red dots indicate vHS at $d_{xz/yz}$ degeneracy point at $M$. FSs in (e) are $87.5$% of their actual sizes.

Figure 4

DOS (a) and static susceptibility (b) of $V$-renormalized band structure at $E_{\mathrm{vH}}=0$ and corresponding degeneracy-splitting energy ${\mathrm{\Delta }}_M$ (c). Black lines in (a)–(c) for TB fit $t_x^{14}=305\mathrm{meV}$ and blue line for $t_x^{14}=100\mathrm{meV}$. (d) ${\mathrm{\Delta }}_M$ vs $V$ around vHS for $t_x^{14}=100\mathrm{meV}$; curves from bottom to top are for $V_0/V=0.6,0.8,0.9,1.0,1.1$, and $1.2$. The dashed black vertical line in (d) indicates where the vHS coincides with the Fermi level.

Figure 5

Band dispersion in units of eV (left panel) and FS (right panel) in a $d$-wave nematic state. (a) and (b) $V=0.763\mathrm{eV}, E_{\mathrm{vH}}=0$, and $V_0=1.3\mathrm{eV}$. (c) and (d) $V=0.73\mathrm{eV}, E_{\mathrm{vH}}=25\mathrm{meV}$, and $V_0=1.34\mathrm{eV}$. (e) and (f) $V=0.75$ eV and $V_0=1.325\mathrm{eV}$ at 1% electron doping. (g) and (h) With SOC ${\lambda }_{\mathrm{soc}}=28\mathrm{meV}, V=0.763\mathrm{eV}, V_0=1.32\mathrm{eV}$ at 1% electron doping.

Figure 6

(a) Schematics of the electronic structure near $M$ showing the location of the vHS ($E_{\mathrm{vH}}$) before (a) and the degeneracy splitting energy ${\mathrm{\Delta }}_M$ after (b) the nematic transition. (c) ${\mathrm{\Delta }}_M$ as a function of $E_{\mathrm{vH}}$ at fixed $V_0/V=1.8,1.7,1.6,1.4$ from the top line to the bottom line.