  A rapid and anisotropic modification of the Fermi-surface shape can be
associated with abrupt changes in crystalline lattice geometry or in the
magnetic state of a material. In this study we show that such an electronic
topological transition is at the basis of the formation of an unusual
pressure-induced tetragonal ferromagnetic phase in Fe$_{1.08}$Te. Around 2 GPa,
the orthorhombic and incommensurate antiferromagnetic ground-state of
Fe$_{1.08}$Te is transformed upon increasing pressure into a tetragonal
ferromagnetic state via a conventional first-order transition. On the other
hand, an isostructural transition takes place from the paramagnetic
high-temperature state into the ferromagnetic phase as a rare case of a `type
0' transformation with anisotropic properties. Electronic-structure
calculations in combination with electrical resistivity, magnetization, and
x-ray diffraction experiments show that the electronic system of Fe$_{1.08}$Te
is instable with respect to profound topological transitions that can drive
fundamental changes of the lattice anisotropy and the associated magnetic
order.
