Spectral density in a nematic state of iron pnictides

Using cluster-perturbation theory, we calculate the spectral density $A(\mathbf{k},\omega )$ for a nematic phase of models describing pnictide superconductors, where very short-range magnetic correlations choose the ordering vector $(\pi ,0)$ over the equivalent $(0,\pi )$ and thus, break the fourfold rotation symmetry of the underlying lattice without inducing long-range magnetic order. In excellent agreement with angle-resolved photoemission spectroscopy (ARPES), we find that the $yz$ bands at $X$ move to higher energies. When on-site Coulomb repulsion brings the system close to a spin-density wave (SDW) and renormalizes the bandwidth by a factor of $\approx 2$, even small anisotropic couplings of 10–$15\text{meV}$ strongly distort the bands, splitting the formerly degenerate states at $X$ and $Y$ by $\approx 70\text{meV}$ and shifting the $yz$ states at $X$ above the chemical potential. This similarity to the SDW bands is in excellent agreement with ARPES. An important difference to the SDW bands is that the $yz$ bands still cross the Fermi level, again in agreement with experiments. We find that orbital weights near the Fermi surface provide a better characterization than overall orbital densities and orbital polarization.

Figure 1

Schematic illustration of the cluster decomposition into (a) four-site and (b) three-site clusters used for the three- and four-orbital models. Ground-state energies and Green's functions of the small cluster—as connected by thick solid lines—are obtained by exact diagonalization. Clusters are then connected in CPT along the thinner dashed bonds. Within the cluster, AFM (FM) Heisenberg exchange acts between electrons along the $x$ ($y$) bond. (c) Spectral density $A(\mathbf{k},\omega )$ of the noninteracting three-orbital model, Eqs. (1, 2, 3, 4, 5, 6). Solid lines indicate the bands in terms of the pseudocrystal momentum $\tilde{\mathbf{k}}$ and shading indicates the spectral weight in terms of “real” momentum $\mathbf{k}$; the difference is that weight with $xy$ character shifts by $(\pi ,\pi )$. [Note that along $(0,0)$-$(\pi ,\pi )$, the bands with dominant $xz$/$yz$ character contain these two orbitals with identical weight, even though the $yz$ character—here drawn on top—dominates the figures.] (d) $A(\mathbf{k},\omega )$ of the noninteracting four-orbital model. Dashed lines indicate the results for the model that is obtained by removing[24] the $3z^2-r^2$ orbital from the five-orbital model of Ref. 25. Solid lines are for the model used here, where the $x^2-y^2$ orbital is then somewhat removed from the Fermi level by changing $t_{xx}^{33}$ from $-0.02$ to $t_{xx}^{33}=0.03$ and ${\epsilon }_3$ from $-0.22$ to $-0.12$ (notation as in Ref. 25). As in (c), shading indicates the spectral weight for the real momentum instead of the pseudocrystal momentum. In the online version, red refers to $xz$, blue to $yx$, and green to all other orbitals. In all spectra, peaks are broadened by a Lorentzian $\delta /[{(\omega -{\omega }_0)}^2+{\delta }^2]$ with $\delta =0.05$ except for (d), where $\delta =0.025$. All energies are in eV.

Figure 2

(a) Spectral density $A(\mathbf{k},\omega )$ and (b) Fermi surface of the three-orbital model (four-site cluster) with parameters as given for Fig. 1c and an explicit symmetry-breaking $J_{\mathrm{anis}}=0.5\text{eV}$ [see Eq. (7)], but without Coulomb repulsion and Hund's rule coupling. Shadings are for real momentum; lines indicate the noninteracting model in pseudocrystal momentum $\tilde{\mathbf{k}}$.

Figure 3

Spectral density $A(\mathbf{k},\omega )$ of the four-orbital model (three-site cluster) [see Fig. 1d], and an increasing explicit symmetry-breaking term, Eq. (7), of (a) $J_{\mathrm{anis}}=0.2\text{eV}$, (b) $J_{\mathrm{anis}}=0.3\text{eV}$, and (c) $J_{\mathrm{anis}}=0.4\text{eV}$. Coulomb repulsion and Hund's rule coupling are not included. Shadings are for real momentum; solid lines indicate the noninteracting model in pseudocrystal momentum $\tilde{\mathbf{k}}$. In (c), dashed lines are for a noninteracting model with an energy difference $\Delta =0.15\text{eV}$ between the $xz$ and $yz$ orbitals, which was fitted to approximately reproduce the difference between the $X$ and $Y$ points.

Figure 4

Density of states for (a) the three-orbital model with $J_{\mathrm{anis}}=0.5\text{eV}$ and (b) the four-orbital model and $J_{\mathrm{anis}}=0.4\text{eV}$. $U=J_{\mathrm{Hund}}=0$ in both cases.

Figure 5

Spectral density with anisotropic short-range magnetic order and on-site interactions. (a) For the four-orbital model and $U=0.3\text{eV}$, $J_{\mathrm{Hund}}=0.075\text{eV}$ and $J_{\mathrm{anis}}=0.2\text{eV}$. (b) For the three-orbital model with $U=0.6\text{eV}$ ($J_{\mathrm{Hund}}=0.15\text{eV}$) and $J_{\mathrm{anis}}=0.2\text{eV}$ and (c) $U=1.02\text{eV}$ ($J_{\mathrm{Hund}}=0.255\text{eV}$) and $J_{\mathrm{anis}}=0.015\text{eV}$. For these last values of $U$ and $J_{\mathrm{Hund}}$, the three-orbital model is very close to the SDW. (d) FS corresponding to the parameters in (c), it captures the spectral weight within $1\text{meV}$ of the Fermi level; broadening of the spectral weight is consistent with (c). On-site interactions are lower for the four-orbital model, because it is at half filling, and thus Hund's rule moves it closer to a Mott transition, while it partly compensates $U$ away from half filling, as in the three-orbital model. Shadings are for real momentum; lines indicate the noninteracting model in pseudocrystal momentum $\tilde{\mathbf{k}}$.