Interplay of nematic and magnetic orders in FeSe under pressure

We offer an explanation for the recently observed pressure-induced magnetic state in the iron-chalcogenide FeSe based on ab initio estimates for the pressure evolution of the most important Coulomb interaction parameters. We find that an increase of pressure leads to an overall decrease mostly in the nearest-neighbor Coulomb repulsion, which in turn leads to a reduction of the nematic order and the generation of magnetic stripe order. We treat the concomitant effects of band renormalization and the induced interplay of nematic and magnetic order in a self-consistent way and determine the generic topology of the temperature-pressure phase diagram and find qualitative agreement with the experimentally determined phase diagram.

Figure 1

(a, d) Orbital-resolved Fermi surfaces obtained from the spectral function at filling $n=6$ for different values of the NN Coulomb interaction parameters. Renormalizations due to the local interaction are typically small and are neglected here. (a) The Fermi surface of the tight-binding model without additional renormalizations, $\tilde{V}={\tilde{V}}_0=0$. (b, c) The Fermi surface including the self-consistent band renormalization ${\chi }_{\text{br}}^{\mu \nu }$ due to NN Coulomb repulsion, (b) $\tilde{V}=0.35\mathrm{eV}$, (c) $\tilde{V}=0.74\mathrm{eV}$. (d) The Fermi surface in the nematic state stabilized by Fermi surface renormalization and a self-consistently $C_4$ symmetry-breaking contribution, $\tilde{V}=0.74\mathrm{eV},{\tilde{V}}_0/\tilde{V}=1.8$. (e)–(f) Cuts through the electronic spectrum as extracted from the spectral function in a symmetric interval $[-0.1,0.1]\mathrm{eV}$ around the Fermi level. The renormalized, tetragonal bands corresponding to the Fermi surface in (c) are shown in black as a reference. The colored curves show the orbitally resolved bands in the nematic state, corresponding to (d). The $d_{xy}$-dominated states are hardly affected by the nematic order, while the $d_{xz}$ and $d_{yz}$ states experience a sizable shift. The deformation of the bands by the nematic order parameter is effective mostly close to the Fermi level. We note that the nematic order parameter does not only shift the states at $X$ and $Y$, but also induces a splitting at the $\mathrm{\Gamma }$ point.

Figure 2

(a)–(c) Phase diagrams in NN Coulomb interaction strength $\alpha \tilde{V}$ vs. temperature $T/\alpha$ parameter space for the FeSe model. The red shaded region denotes the nematic state (N), while the blue shaded region corresponds to the magnetic stripe state (SDW). The coexistence of nematic order and magnetic order is indicated by magenta color. The points indicate the parameters where self-consistent calculations have been carried out. We fixed the ratio of ${\tilde{V}}_0/\tilde{V}=1.8$ generating a splitting of about 50 meV of the degenerate $d_{xz}$ and $d_{yz}$ orbitals at the $M$ point (with respect to the 2-Fe BZ) in the nematic state. For the displayed phase diagrams we fix $\alpha U=1.40$ eV and study the impact of the rescaled Hund's coupling $\alpha J$ on the phase diagram, where in (a) $\alpha J=0.325$ eV, (b) $\alpha J=0.350$ eV, and (c) $\alpha J=0.375$ eV. The black curves show the integrated density of states in an energy range $[-25,+25]$ meV to around the Fermi level with $\alpha T=2$ meV as a function of $V$ to obtain a naive estimate of the superconducting $T_{\mathrm{c}}$.

Figure 3

(a)–(c) Reconstructed Fermi surfaces for (a) $\alpha \tilde{V}=0.77$ eV, (b) $\alpha \tilde{V}=0.73$ eV, and (c) $\alpha \tilde{V}=0.69$ eV for $\alpha U=1.40$ eV and $\alpha J=0.350$ eV at $\alpha T=2$ meV, corresponding to the states in the phase diagram shown in Fig. 2.

Figure 4

(a) Pressure-induced renormalization of onsite $\left(U,U^{\prime },J\right)$, NN $\left(V\right)$, and NNN $\left(V^{\prime }\right)$ couplings extracted from DFT calculations relative to the $p=0$ MPa value. The only coupling that shows a clear upward trend under application of pressure is Hund's coupling $J$. The intra- and interorbital repulsions $U$ and $U^{\prime }$ as well as the longer-ranged repulsion $V$ and $V^{\prime }$ decrease with pressure. The longer-ranged interactions are clearly more affected and display changes between $\sim 10$ % $\left(V\right)$ and $\sim 15$ % $\left(V^{\prime }\right)$. (b) Pressure dependence of couplings rescaled with a rough estimate of the hopping renormalization from intralayer hoppings, ${\alpha }_{\parallel }^{-1}$. (c) Estimates of the hopping renormalizations for inter- and intralayer hoppings, ${\alpha }_{\perp }^{-1}$ and ${\alpha }_{\parallel }^{-1}$ as a function of pressure.

Figure 5

(a)–(c) Electronic spectral weight along the high-symmetry cut $\mathrm{\Gamma }\text{−}X\text{−}M\text{−}\mathrm{\Gamma }$ in momentum space in a symmetric low-energy window of width $0.2\mathrm{eV}$ around the Fermi level. Here $U,J=0$ and ${\tilde{V}}_0=0$. As the interaction strength $\tilde{V}$ is increased from (a) $\tilde{V}=0.69$ eV over (b) $\tilde{V}=0.73$ eV to (c) $\tilde{V}=0.77$ eV, both electron and hole pockets decrease in size. (d)–(f) The corresponding Fermi surfaces, where for clarity we folded the electron pockets around $X$ onto the central hole pockets with the folding-vector $(\pi ,0)$ to illustrate the varying degree of $(\pi ,0)$ nesting. The same conclusions are obtained for the electron pocket at $Y$ in the nonnematic phase. The nesting is close to optimal for (e). In the cases (d) and (f) nesting is in fact suppressed by matrix-element effects.

Figure 6

The static RPA susceptibility ${\chi }^{\mathrm{RPA}}(\omega =0,\mathbf{q})$ with momentum transfer $\mathbf{q}=(\pi ,0)$ as a function of different interaction strength $\tilde{V}$ for (a) $U=1.30$ eV and (b) $U=1.40$ eV and increasing Hund's coupling (from bottom to top) $J=0.1,0.2,0.3,0.32,0.324\mathrm{eV}$ at temperature $T=2$ meV. There is a clear enhancement of $(\pi ,0)$ spin fluctuations in the vicinity of the point where the band renormalization pushes the van Hove singularity through the Fermi level.