  Uniaxial pressure applied along an Ru-Ru bond direction induces an elliptical
distortion of the largest Fermi surface of Sr$_2$RuO$_4$, eventually causing a
Fermi surface topological transition, also known as a Lifshitz transition, into
an open Fermi surface. There are various anomalies in low-temperature
properties associated with this transition, including maxima in the
superconducting critical temperature and in resistivity. In the present paper,
we report new measurements, employing new uniaxial stress apparatus and new
measurements of the low-temperature elastic moduli, of the strain at which this
Lifshitz transition occurs: a longitudinal strain $\varepsilon_{xx}$ of
$(-0.44\pm0.06)\cdot10^{-2}$, which corresponds to a B$_{1g}$ strain
$\varepsilon_{xx} - \varepsilon_{yy}$ of $(-0.66\pm0.09)\cdot10^{-2}$. This is
considerably smaller than the strain corresponding to a Lifshitz transition in
density functional theory calculations, even if the spin-orbit coupling is
taken into account. Using dynamical mean-field theory we show that electronic
correlations reduce the critical strain. It turns out that the orbital
anisotropy of the local Coulomb interaction on the Ru site is furthermore
important to bring this critical strain close to the experimental number, and
thus well into the experimentally accessible range of strains.
