  A common feature of topological insulators is that they are characterized by
topologically invariant quantity such as the Chern number and the
$\mathbb{Z}_2$ index. This quantity distinguishes a nontrivial topological
system from a trivial one. A topological phase transition may occur when there
are two topologically distinct phases, and it is usually defined by a gap
closing point where the topologically invariant quantity is ill-defined. In
this paper, we show that the magnon bands in the strained (distorted)
kagome-lattice ferromagnets realize an example of a topological magnon phase
transition in the realistic parameter regime of the system. When spin-orbit
coupling (SOC) is neglected (i.e. no Dzyaloshinskii-Moriya interaction), we
show that all three magnon branches are dispersive with no flat band, and there
exists a critical point where tilted Dirac and semi-Dirac point coexist in the
magnon spectra. The critical point separates two gapless magnon phases as
opposed to the usual phase transition. Upon the inclusion of SOC, we realize a
topological magnon phase transition point at the critical strain
$\delta_c=\frac{1}{2}\big[ 1-(D/J)^2\big]$, where $D$ and $J$ denote the
perturbative SOC and the Heisenberg spin exchange interaction respectively. It
separates two distinct topological magnon phases with different Chern numbers
for $\delta<\delta_c$ and for $\delta>\delta_c$. The associated anomalous
thermal Hall conductivity develops an abrupt change at $\delta_c$, due to the
divergence of the Berry curvature in momentum space. The proposed topological
magnon phase transition is experimentally feasible by applying external
perturbations such as uniaxial strain or pressure.
