  The Fermi surface topology plays an important role in the macroscopic
properties of metals. It can be particularly sensitive to electron correlation,
which appears to be especially significant for the weak itinerant ferromagnet
ZrZn$_{2}$. Here, we look at the differences in the predicted Fermi surface
sheets of this metallic compound in its paramagnetic phase for both density
functional theory (DFT) and the combination of DFT with dynamical mean field
theory (DFT+DMFT). The theoretical spectral functions evaluated at the Fermi
level were used along with calculations of the electron-positron momentum
density (also known as the two-photon momenutm density) in $k$-space to provide
insights into the origin of certain features of the Fermi surface topology. We
compare this two photon momentum density to that extracted from the positron
annihilation experimental data (Phys. Rev. Lett. 92, 107003 (2004)). The
DFT+DMFT densities are in better agreement with the experiment than the DFT,
particularly with regard to the flat bands around the $L$ and $W$ high symmetry
points. The experimental neck around $L$, which relates to a van Hove
singularity, is present in DFT+DMFT but not in the DFT. We find that these flat
bands, and as such the Fermi surface topology, are sensitive to the many body
electron correlation description, and show that the positron annihilation
technique is able to probe this. This description is significant for the
observed behavior such as the Lifshiftz transition around the quantum critical
point.
