## 2 Results  
### Single-Particle Properties Near the Fermi Surface  
In addition to an ARPES measurements of the Fermi surface in 2014 [35], a new high-resolution laser ARPES measurement was reported [32], which provides an excellent benchmark of the theory. In [17], we compared the Fermi surface; here, we present a more detailed benchmark. * In the unstrained case, DFT yields a good Fermi surface [32], but Q\(SGW\) performs a little better. Q\(SGW\) and Q\(SGW^{++}\) are very similar (left panel, Figure 1). In the vicinity of the \(\theta=45\) degree line connecting \(\Gamma\) and X, the inner (\(\beta\)) pocket makes a smooth transition from \(xz\) (green) to \(xy\) (red) at the 45 degree mark, to the \(yz\) (blue) character (middle panel, Figure 1). Thus, \(xy\) and \(xz\),\(yz\) exchange roles on this line. The nesting vector along this line give rise to peaks in \(\chi^{m}\) at \(q=(\pm 0.3,\pm 0.3)\), which mainly drive superconductivity. * The tendency for Q\(SGW\) to yield smaller \(v_{F}\) than DFT is typical [31]; and so is further renormalisation from spin fluctuations, computed via DMFT. It is noteworthy that \(v_{F}\) computed in Q\(SGW^{++}\) (hexagons) is in very good agreement with ARPES data (blue circles), including the variation with \(\theta\). * Under strain, the \(xy\) symmetry is broken: the two points MGin the middle panel of Figure 1 become different. One point contracts, and the other widens, touching the boundary at a critical strain and causing a topological (Lifshitz) transition. Q\(SGW\) and DMT capture this transition, but Q\(SGW\) accurately captures the critical strain (\(\epsilon_{x}=0.6\%\)), whereas DFT gets it severely wrong [17]. * Spin-orbit modification of the band structure is poorly described in DFT; Reference [32] referred to \"correlation enhancements\" to it. Q\(SGW\), however, describes the splitting very well (\(\sim\)90 meV), in good agreement with an estimate of 100 meV [32], revised downward from the 2014 estimate of 130 \(\pm\) 30 meV [35]. Figure 2a shows how the orbitally resolved electronic masses and single-particle scattering rates evolve with strain \(\epsilon_{x}\). The single-site DMFT \(\mathrm{Im}\,\Sigma(i\omega)\) is fit to a fourth order polynomial in \(i\omega\) for low energies (first six Matsubara points at \(\beta=40\,\mathrm{eV}^{-1}=290\,\mathrm{K}\)) [36]. The mass enhancement, related to the coefficient \(s_{1}\) of the linear term in the expansion  
Figure 1: Dynamical mean field theory (DMFT) spectral function in Sr\({}_{2}\)RuO\({}_{4}\) at 145 K (left). Yellow lines are Q\(SGW\) bands; green lines extract the peak in the Q\(SGW^{++}\) spectral functions near the Fermi level. The middle panel shows the Q\(SGW\) Fermi surface, showing the inner (\(\beta\)), middle (\(\gamma\)), and outer (a) pockets resolved by the orbital character (red = \(xy\), green = \(xz\), blue = \(yz\)). Point MGis midway between the M and \(\Gamma\) points. Q\(SGW\), Q\(SGW^{++}\), and ARPESermi surfaces are all very similar, with Q\(SGW^{++}\) being slightly closer to the ARPES measurement. The right figure shows the Fermi velocities \(v_{F}\) (\(10^{5}\,\mathrm{m/s}\)) of the \(\beta\) and (\(\gamma\)) pockets, on a line connecting \(q=0\) to a zone boundary point, whose direction is given by \((\cos(\theta),\sin(\theta))\). \(m_{\text{DMFT}}/m_{\text{QSGW}}=1+|s_{1}|\)[37] and the intercept \(|s_{0}|=\Gamma m_{\text{DMFT}}/m_{\text{QSGW}}\) with \(m_{\text{DMFT}}/m_{\text{QSGW}}=Z^{-1}\), is resolved in different intra-orbital channels. Both the masses and \(\Gamma\) are orbital-dependent, and this differentiation is a signature of a Hund's metal [21]. Electrons in the \(d_{xy}\) orbital become heavier, while the \(d_{xz,yz}\) electrons become lighter with \(\epsilon_{x}\). Beyond a critical \(\epsilon_{x}\sim 0.6\%\), \(m_{xy,\text{DMFT}}/m_{xy,\text{QSGW}}\) becomes heavier than \(m_{xz,\text{DMFT}}/m_{xz,\text{QSGW}}\) (see Figure 2b).