  Material realization of the non-Abelian Kitaev spin liquid phase - an example
of Ising topological order (ITO) - has been the subject of intense research in
recent years. The $4d$ honeycomb Mott insulator $\alpha$-RuCl$_3$ has emerged
as a leading candidate, as it enters a field-induced magnetically disordered
state where a half-integer quantized thermal Hall conductivity $\kappa_{xy}$
was reported. Further, a recent report of a sign change in the quantized
$\kappa_{xy}$ across a certain crystallographic direction is strong evidence
for a topological phase transition between two ITOs with opposite Chern
numbers. Although this is a fascinating result, independent verification
remains elusive, and one may ask if there is a thermodynamic quantity sensitive
to the phase transition. Here we propose that the magnetotropic coefficient $k$
under in-plane magnetic fields would serve such a purpose. We report a singular
feature in $k$ that indicates a topological phase transition across the
$\hat{b}$-axis where ITO is prohibited by a $C_2$ symmetry. If the transition
in $\alpha$-RuCl$_3$ is indeed a direct transition between ITOs, then this
feature in $k$ should be observable.
