  Although the overwhelming majority of natural processes occurs far from the
equilibrium, general theoretical approaches to non-equilibrium phase
transitions remain scarce. Recent breakthroughs introducing description of open
dissipative systems in terms of non-Hermitian quantum mechanics allowed to
identify a class of non-equilibrium phase transitions associated with the loss
of combined parity (reflection) and time-reversal symmetries. Here we report
that time evolution of a single classical spin (e.g. monodomain ferromagnet)
governed by the Landau-Lifshitz-Gilbert-Slonczewski equation in absence of
higher-order anisotropy terms is described by a M\"{o}bius transformation in
complex stereographic coordinates. We identify the \textit{parity-time}
symmetry-breaking phase transition occurring in spin-transfer torque-driven
linear spin systems as a transition between hyperbolic and loxodromic classes
of M\"{o}bius transformations, with the critical point of the transition
corresponding to the parabolic transformation. This establishes the
understanding of non-equilibrium phase transitions as topological transitions
in configuration space.
