Multipolar nematic state of nonmagnetic FeSe based on $\mathrm{DFT}+U$

Clarifying the origin of nematic state in FeSe is one of urgent problems in the field of iron-based superconductivity. Motivated by the discovery of a nematic solution in the density-functional theory implemented by on-site Coulomb interaction ($\mathrm{DFT}+U$) [npj Quantum Mater. 5, 50 (2020)], we reexamine the $U$ dependence of electronic states in the nonmagnetic normal state of FeSe and perform full multipolar analyses for the nematic state. We find that with increasing $U$ the normal state experiences a topological change in the Fermi surfaces before the emergence of a nematic ground state. The resulting nematic ground state is a multipolar state having both antiferrohexadecapoles in the $E$ representation and ferromultipoles in the $B_2$ representation on each Fe site. Cooperative coupling between the $E$ and the $B_2$ multipoles in the local coordinate with the $D_{2d}$ point group will play an important role in the formation of the $d_{xz},d_{yz}$ orbital-splitting nematic state not only in FeSe, but also in other iron pnictides.

Figure 1

$U$ dependence of pDOS ${\rho }_m^d\left({\varepsilon }_{\mathrm{F}}\right)$. The inset shows the 3h-FSs for $U=0\mathrm{eV}$ (left center) and the 2h-FSs for $U=3\mathrm{eV}$ (right top).

Figure 2

(a) Energy differences of the total energy $\mathrm{\Delta }{E}_{\mathrm{tot}}$ (left axis), and kinetic and potential energies $\mathrm{\Delta }{T}_s,\mathrm{\Delta }{U}_{\mathrm{pot}}$ (right axis) and (b) $n_m^d$. (c)–(f) The obtained band energy ${\varepsilon }_{\mathit{k}n}-{\varepsilon }_{\mathrm{F}}$ around ${\varepsilon }_{\mathrm{F}}$ for $U=3\mathrm{to}4\mathrm{eV}$.

Figure 3

The quadrupole and hexadecapole moments (a) and (b) $O_{{\mathrm{\Gamma }}_{\gamma }}$ and (c) and (d) $Q_{4{\mathrm{\Gamma }}_{\gamma }}$ on Fe1 [(a) and (c)] and Fe2 [(b) and (d)] sites as a function of $U$. In the insets of (a) and (b), atom sites in the unit cell are depicted with $x,y$ axes.

Figure 4

Band structures for (a)–(c) normal state ($U=3.3\mathrm{eV}$) and (d)–(f) nematic state ($U=3.4\mathrm{eV}$) with the orbital weight of (a) and (d) $d_{xy}$, (b) and (e) $d_{xz}$(green), $d_{yz}$ (blue), and (c) and (f) $d_{x^2-y^2}$ along the high-symmetry line in the BZ, where ${\mathrm{M}}_x\left[{\mathrm{M}}_y\right]=(\frac{\pi }{a},\frac{\pi }{a},0])\left[(\frac{\pi }{a},-\frac{\pi }{a},0)\right]$ and $A_x\left[A_y\right]=(\frac{\pi }{a},\frac{\pi }{a},\frac{\pi }{c})\left[(\frac{\pi }{a},-\frac{\pi }{a},\frac{\pi }{c})\right]$ and $a,c$ are the lattice constants.