Orbital order in FeSe: The case for vertex renormalization

We study the structure of the orbital order $\mathrm{\Gamma }=〈d_{xz}^{†}{d}_{xz}-d_{yz}^{†}{d}_{yz}〉$ in FeSe in light of recent scanning tunneling microscopy and angle-resolved photoemission spectroscopy (ARPES) data, which detect the shapes of the hole and electron pockets in the nematic phase. The geometry of the pockets indicates that the sign of $\mathrm{\Gamma }$ is different between the hole and electron pockets (${\mathrm{\Gamma }}_h$ and ${\mathrm{\Gamma }}_e$). We argue that this sign change cannot be reproduced if one solves for the orbital order within the mean-field approximation, as the mean-field analysis yields either no orbital order, or order with the same sign of ${\mathrm{\Gamma }}_e$ and ${\mathrm{\Gamma }}_h$. We argue that another solution with the opposite signs of ${\mathrm{\Gamma }}_e$ and ${\mathrm{\Gamma }}_h$ emerges if we include the renormalizations of the vertices in the $d$-wave orbital channel. We show that the ratio $|{\mathrm{\Gamma }}_e/{\mathrm{\Gamma }}_h|$ is of order one, independent of the strength of the interaction. We also compute the temperature variation of the energy of $d_{xz}$ and $d_{yz}$ orbitals at the center of electron pockets and compare the results with ARPES data.

Figure 1

(a) Hartree and Fock self-energy diagrams. (b) Examples of the interaction terms which contribute to Hartree-Fock self-energies. The $U_5$ terms in the first row also act on the hole pockets (${\psi }_5,{\psi }_6$). Each diagram has symmetry equivalents (${\psi }_1\leftrightarrow {\psi }_3$, ${\psi }_5\leftrightarrow {\psi }_6$). The self-energy beyond the mean-field approximation has been computed using dressed interactions, which we obtained using the pRG scheme. In a direct perturbation theory, this amounts to summing up an infinite series of self-energy diagrams, including RPA and Aslamazov-Larkin diagrams.

Figure 2

(a) and (b) The pRG flow of the couplings $U_a$ and $U_b$ for the case when the bare $U_a^{\left(0\right)}=U_b^{\left(0\right)}=U_0=5J-U$ is positive in (a) and negative in (b) (we set $J/U=0.3$ and 0.1, respectively). (c) and (d) The flow of the couplings $U_M$ and $U_{\mathrm{\Gamma }}$ in Eq. (3). The parameter $L=log\frac{W}{E}$, where $W$ is of order bandwidth and $E$ is the running energy. The larger $L$ is, the more high-energy states are integrated out. We used $m_{h}U/\left(4\pi \right)=0.35$, where $m_h$ is the mass of the dispersion near the hole pocket.

Figure 3

The flow of the dimensionless couplings ${\lambda }^{++}$ in the sign-preserving $d^{++}$ channel (green) and ${\lambda }^{+-}$ in the sign-changing $d^{+-}$ channel (red). Notations are as in Fig. 2. (a) and (b) The flow for the case $U_0=5J-U>0$ for two values of the parameter $\gamma =A_e/A_h$ (see text). (c) and (d) The same for $U_0<0$. The sign-changing $d^{+-}$ channel becomes dominant once $U_b$ changes sign near $L=2$. For $\gamma \ne 1$, the couplings jump by finite values when $U_b$ passes through zero.