  We present results of a theoretical study of a prototypical weak ferromagnet
ZrZn$_2$. We use the density-functional theory (DFT)+dynamical mean-field
theory (DMFT) method to study the electronic and local magnetic properties. The
obtained DFT+DMFT electronic self-energies are Fermi-liquid like, indicating a
small effective mass enhancement of the Zr $4d$ states $m^*/m\sim 1.1 - 1.3$
accompanied by partly formed local moments within the electronic states of
$t_{2g}$ symmetry. The effect of electronic interaction is shown to be
essential for determining the correct topology of some of the Fermi surface
sheets. To study in detail the pressure dependence of the Curie temperature
$T_{\rm C}$ and corresponding pressure-induced quantum phase transition, we
consider an effective single-band model, constructed using the Zr $4d$
contribution to the total density of states. The model is studied within static
and dynamic mean-field theory, as well as spin-fermion approach. We show that
the spin-fermion approach yields the temperature dependence of susceptibility
at ambient pressure and the pressure dependence $T_{\rm C}(p)$, including the
first-order quantum phase transition at $p\approx 1.7$~GPa, comparable well
with the experimental data.
