---
author:
- Mengshu Liu
- 'Leland W. Harriger'
- Huiqian Luo
- Meng Wang
- 'R. A. Ewings'
- 'T. Guidi'
- Hyowon Park
- Kristjan Haule
- Gabriel Kotliar
- 'S. M. Hayden'
- Pengcheng Dai
title: 'Nature of magnetic excitations in superconducting BaFe$_{1.9}$Ni$_{0.1}$As$_{2}$'
---

Our BaFe$_{2-x}$Ni$_{x}$As$_{2}$ electron-doped samples were grown using self flux method as described before [@yanchao]. Since the electronic and superconducting properties of Co and Ni-doping of BaFe$_2$As$_2$ near optimal superconducting transition temperatures are almost identical [@budko], we chose to study spin excitations in optimally doped BaFe$_{1.9}$Ni$_{0.1}$As$_{2}$ with $T_c=20$ K. To further compare spin excitations in BaFe$_2$As$_2$ and BaFe$_{1.9}$Ni$_{0.1}$As$_{2}$, we show in Figure 1 spin excitations of these two materials at different energies. Consistent with data shown in Figs. 2-4 of the main text, we see that the effect of electron-doping is to mostly modify spin excitations below 80 meV.

![ Constant-energy images of the spin excitations as a function of increasing energy for BaFe$_{1.9}$Ni$_{0.1}$As$_{2}$ and BaFe$_2$As$_2$ in units of mbarns/sr/meV/f.u. (a) $E=33\pm 3$, (b) $43\pm 3$, (c) $60\pm 10$, (d) $81\pm 10$, (e) $113\pm 10$, (f) $135\pm 10$, (g) $157\pm 10$, and (h) $214\pm 10$ meV. ](su1)

Our theoretical method for computing the magnetic excitation spectrum employs abinitio full potential DFT+DMFT method, as implemented in Ref. [@Haule], which is based on the commercial DFT code of Wien2k [@Wien2K]. The DMFT method requires solution of the generalized quantum impurity problem, which is here solved by the numerically exact continuous-time quantum Monte Carlo method (CTQMC) [@Haule:07; @Werner:06]. The Coulomb interaction matrix for electrons on iron atom was determined by the self-consistent GW method in Ref. [@Kutepov:10], giving $U=5\,$eV and $J=0.8\,$eV for the local basis functions within the all electron approach employed in our DFT+DMFT method.

![ The Feynman diagrams for the Bethe-Salpeter equation. It relates the two-particle Green’s function ($\chi$) with the polarization ($\chi^{0}$) and the local irreducible vertex function ($\Gamma_{loc}^{irr}$). The non-local two-particle Green’s function is obtained by replacing the local propagator by the non-local propagator. \[fig:BSE\]](diagram)

The dynamical magnetic susceptibility $\chi(\textbf{q},\omega)$ is computed from the *ab initio* perspective by extracting the two-particle vertex functions of DFT+DMFT solution $\Gamma_{loc}^{irr}$. The polarization bubble $\chi^{0}$ is computed from the fully interacting one particle Greens function. The full susceptibility is computed from $\chi^{0}$ and the two-particle irreducible vertex function $\Gamma_{loc}^{irr}$, which is assumed to be local in the same basis in which the DMFT self-energy is local, implemented here by the projector to the muffin-tin sphere [@Haule]. In order to extract $\Gamma_{loc}^{irr}$, we employ the Bethe-Salpeter equation (see Fig. \[fig:BSE\]) which relates the local two-particle Green’s function ($\chi_{loc}$), sampled by CTQMC, with both the local polarization function ($\chi_{loc}^{0}$) and $\Gamma_{loc}^{irr}$: $$\Gamma_{loc{\alpha_{1}\sigma_{1},\alpha_{2}\sigma_{2}\atop
    \alpha_{3}\sigma_{3},\alpha_{4}\sigma_{4}}}^{irr}(i\nu,i\nu^{\prime})_{i\omega}=\frac{1}{T}[(\chi_{loc}^{0})_{i\omega}^{-1}-\chi_{loc}^{-1}].$$ $\Gamma_{loc}^{irr}$ depends on three Matsubara frequencies ($i\nu$, $i\nu^{\prime}$; $i\omega$), and both the spin ($\sigma_{1-4}$) and the orbital ($\alpha_{1-4}$) indices, which run over $3d$ states on the iron atom. $T$ is the temperature.

Once the irreducible vertex $\Gamma_{loc}^{irr}$ is obtained, the momentum dependent two-particle Green’s function is constructed again using the Bethe-Salpeter equation (Fig. \[fig:BSE\]) by replacing the local polarization function $\chi_{loc}^{0}$ by the non-local one $\chi_{\textbf{q},i\omega}^{0}$: $$\chi_{{\alpha_{1}\sigma_{1},\alpha_{2}\sigma_{2}\atop
    \alpha_{3}\sigma_{3},\alpha_{4}\sigma_{4}}}(i\nu,i\nu^{\prime})_{\textbf{q},i\omega}=[(\chi^{0})_{\textbf{q},i\omega}^{-1}-T\cdot\Gamma_{loc}^{irr}]^{-1}.
\label{eq:BSE_imag}$$ Finally, the dynamic magnetic susceptibility $\chi(\textbf{q},i\omega)$ is obtained by closing the two particle green’s function with the magnetic moment $\mu=\mu_B(\textbf{L}+2\textbf{S})$ vertex, and summing over all internal degrees of freedom, i.e., orbitals ($\alpha_{1-4}$), spins ($\sigma_{1-4}$) and frequencies ($i\nu$,$i\nu^{\prime}$), on the four external legs $$\chi(\textbf{q},i\omega)=T\sum_{i\nu,i\nu^{\prime}}\sum_{{\alpha_{1}\alpha_{2}\atop\alpha_{3}\alpha_{4}}}\sum_{{\sigma_{1}\sigma_{2}\atop
    \sigma_{3}\sigma_{4}}}\mu_{{\alpha_{1}\sigma_{1}\atop\alpha_{3}\sigma_{3}}}^{z}
\mu_{{\alpha_{2}\sigma_{2}\atop\alpha_{4}\sigma_{4}}}^{z} \;\chi_{{\alpha_{1}\sigma_{1},\alpha_{2}\sigma_{2}\atop
    \alpha_{3}\sigma_{3},\alpha_{4}\sigma_{4}}}(i\nu,i\nu^{\prime})_{\textbf{q},i\omega}
\label{eq:chi}$$

We note that the same abinitio methodology which is here used to compute the magnetic excitation spectra, was previously shown to describe the photoemission, the optical spectra and the magnetic moments of this material [@Yin:11] in excellent agreement with experiment.

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