  The Fe sublattice in the compound ZrFe$_4$Si$_2$ features geometrical
frustration and quasi-one-dimensionality. We therefore investigated the
magnetic behavior in ZrFe$_4$Si$_2$ and its evolution upon substituting Ge for
Si and under the application of hydrostatic pressure using structural,
magnetic, thermodynamic, and electrical-transport probes. Magnetic measurements
reveal that ZrFe$_4$Si$_2$ holds paramagnetic Fe moments with an effective
moment $\mu_{\rm eff}= 2.18~\mu_{B}$. At low temperatures the compound shows a
weak short-range magnetic order below 6 K. Our studies demonstrate that
substituting Ge for Si increases the unit-cell volume and stabilizes the
short-range order into a long-range spin-density wave type magnetic order. On
the other hand, hydrostatic pressure studies using electrical-resistivity
measurements on ZrFe$_4$(Si$_{0.88}$Ge$_{0.12}$)$_2$ indicate a continuous
suppression of the magnetic ordering upon increasing pressure. Therefore, our
combined chemical substitution and hydrostatic pressure studies suggest the
existence of a lattice-volume-controlled quantum critical point in
ZrFe$_4$Si$_2$.
