## 2 Results
### Spin Fluctuations: Incommensurability and Coherence  
\(\chi(\mathbf{q},\omega)\) is computed from the momentum-dependent Bethe-Salpeter Equation (1) in the magnetic channel. \[\chi^{\text{m}}_{ \begin{subarray}{c}\mathbf{z}_{1}\
eq 2\\ \mathbf{z}_{3}\
eq 4\end{subarray} }(iv,iv^{\prime})_{\mathbf{q},i\omega}=[( \chi^{0})^{-1}_{\mathbf{q},i\omega}-\Gamma^{irr,m}_{loc}]^{-1}_{ \begin{subarray} {c}\mathbf{z}_{1}\
eq 2\\ \mathbf{z}_{3}\
eq 4\end{subarray} }(iv,iv^{\prime})_{\mathbf{q},i\omega}. \tag{1}\]  
\(\chi^{0}\) is the non-local (\(k\)-dependent) polarisation bubble computed from single-particle QS\(GW\) Green's functions dressed by the local DMFT self-energy, and \(\Gamma\) is the local irreducible two-particle vertex function computed in the magnetic channel. \(\Gamma\) is a function of two fermionic frequencies \(\
u\) and \(\
u^{\prime}\) and the bosonic frequency \(\omega\). \(\chi(\mathbf{q},i\omega)\) is computed by closing \(\chi^{\text{m}(d)}_{ \begin{subarray}{c}\mathbf{z}_{1}\
eq 2\\ \mathbf{z}_{1}\
eq 2\end{subarray} }(iv,iv^{\prime})_{\mathbf{q},i\omega}\) with spin bare vertex \(\gamma\) and summing over frequencies (\(iv\),\(iv^{\prime}\)) and orbitals (\(\mathbf{z}_{1}\),\(\mathbf{z}\)). We compute the real part of the static susceptibility \(\text{Re}\chi(\mathbf{q},i\omega=0)\) and resolve it in different inter- and intra-orbital channels to develop a systematic understanding of which orbitals dominate the spin susceptibilities at different q-vectors. In the vicinity of the ferromagnetic (FM) vector \(\mathbf{q}^{\text{FM}}=(0,0,0)\), \(\chi\) is dominated by the intra-orbital fluctuations in the \(d_{xy}\) channel (Figure 3a), while at the incommensurate (IC) vector \(\mathbf{q}^{\text{IC}}=(0.3,0.3,0)\) (we use units \(2\pi/a\) throughout), the three orbitals contribute almost equally. The antiferromagnetic (AFM) vector \(\mathbf{q}^{\text{AFM}}=(0.5,0.5,0)\) is fully gapped. When strain is applied, we find that the IC peak rapidly increases, and \(d_{xy}\) emerges as the leading component of total spin susceptibilities along all high symmetry directions (see Figure 3b-e). This is consistent with the fact that under strain, \(d_{xy}\) becomes the most strongly correlated orbital. Nevertheless, the AFM vector remains fully gapped for strains up to \(\epsilon_{x}=2.4\%\). We compute both the real and imaginary parts of spin and charge susceptibilities by solving the BSE in the respective channels. These equations are solved in the Matsubara representation with local dynamic vertex functions (which are  
Figure 2: Effective masses and scattering rates: (**a**) The mass enhancement factors in DMFT (relative to the QSGW mass) are plotted in Ru-\(d_{xy,yz,xz}\) channels. While the unstrained compound finds the heaviest electron mass for the electrons in the \(d_{xz}\) orbital, under strain, the \(d_{xy}\) mass becomes the heaviest. (**b**) We show the relative DMFT mass enhancement for the \(\text{d}_{xy}\) orbital in comparison to \(\text{d}_{xz}\) for all temperatures. (**c**) Scattering rates \(\Gamma\) are orbitally anisotropic, but under strain, they decrease in all orbital channels. For very large strains, the system becomes a better Fermi liquid metal; nevertheless, the orbital anisotropy, which is a typical signature of [PERSON]’s metals, survives for the entire range of strain. functions of three Matsubara frequencies) and the non-local polarisation bubble, which also has the Matsubara frequencies. After summing over all internal Fermionic Matsubara frequencies and orbital indices, we are left with \(\chi(\mathbf{q},i\omega)\). Further, it needs to be analytically continued to real bosonic frequencies. One way is to analytically continue \(\chi(i\omega)\) at each momentum, which is tremendously expensive. To understand the precise nature of the spin fluctuations at finite energies, it is imperative in this work that we extract \(\mathrm{Im}\,\chi(\mathbf{q},\omega)\) for finite \(\omega\).