  In this article, we study non-analyticities or cusps near a topological phase
transition driven by changes of global topologies rather than by the formation
of conventional (local) order. The particular phase transitions between
topologically ordered superfluids and non-topological superfluids studied in
this article are always accompanied by the emergence or disappearance of
Majorana boundary modes, while at the critical point these boundary modes are
liberated and evolve into the bulk. Proliferation of massless Majorana fermions
at the transition leaves unique thermodynamic signatures. For a very broad
class of isotropic superfluids or superconductors (with or without time
reversal symmetry), the resultant transitions in $d$ spatial dimensions will be
of $(d+1)$th order. The $(d+1)$th order derivatives of the grand potential are
either discontinuous in even dimensions or logarithmically divergent in odd
dimensions. Bulk quantities such as compressibility can develop a cusp along
with the change of topology of the superfluids. We compute quantitatively the
cusp structures and further investigate the effect of interactions induced by
massive fluctuation fields. These cusps are direct manifestations of Majorana
fermions and their interactions, and can be potentially considered as an
alternative approach for detecting these puzzling particles.
