Nematic state stabilized by off-site Coulomb interaction in iron-based superconductors

Using a variational Monte Carlo method, we investigate the nematic state in iron-based superconductors based on the three-band Hubbard model. Our results demonstrate that the nematic state, formed by introducing an anisotropic hopping order into the projected wave function, can arise in the underdoped regime by an off-site Coulomb interaction $V$ induced $d$-wave Pomeranchuk instability. An analysis of $V$ in momentum space indicates that the interaction between electrons around $(\pi ,0)$ and $(0,\pi )$ points is crucial for the formation of the nematic state. The resulting anisotropic kinetic energy and spin correlation, as well as unequal occupation of $d_{xz}$ and $d_{yz}$ orbitals, are all suppressed upon electron doping, which are consistent with the doping dependence of intrinsic anisotropy revealed by optical spectrum measurement and angle-resolved photoemission spectroscopy.

Figure 1

Condensation energy $E_{\mathrm{cond}}$ (a) and optimized value of ${\delta }_{\mathrm{var}}$ (b) as a function of electron doping on the $20\times 20, 22\times 22$, and $24\times 24$ lattices.

Figure 2

Electron occupation $n\left(k\right)$ in the normal state (${\delta }_{\mathrm{var}}=0.0$) (a) and in the nematic state (${\delta }_{\mathrm{var}}=0.5$) (b) on the $22\times 22$ lattice at $x=0$. The dashed and dotted lines, across which distinct colors manifest a finite change of $n\left(k\right)$, denote the electron and hole Fermi pockets, respectively.

Figure 3

(a) Different energy contributions to the condensation energy as a function of electron doping on the $20\times 20$ lattice. (b) Total energy change as a function of ${\delta }_{\mathrm{var}}$ for different values of $V$ on the $20\times 20$ lattice at $x=0.10$. The different parts of Hamiltonian and the value of $V$ are indicated by the shape of the symbol in (a) and (b), respectively.

Figure 4

Momentum (${\mathbf{k}}^{\prime }$) dependence of the interaction $-2[\text{cos}(k_x-k_x^{{}^{\prime }})+\text{cos}(k_y-k_y^{{}^{\prime }})]$ on the $22\times 22$ lattice at $\mathbf{k}=(\pi ,0)$. Notice that the interaction occurs mainly for electrons around the borders of the Brillouin zone and exhibits a twofold symmetry.

Figure 5

Ratio of ${\mathrm{kin}}_x$ and ${\mathrm{kin}}_y$ (a), $n_{xz}/n_{yz}$ (b), and $\langle S_i·S_{i+{\mathbf{a}}_x}\rangle -\langle S_i·S_{i+{\mathbf{a}}_y}\rangle$ (c) as a function of electron doping on the $20\times 20, 22\times 22$, and $24\times 24$ lattices. The kinetic energies ${\mathrm{kin}}_x$ and ${\mathrm{kin}}_y$ are defined in the text.