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In this article, we study non-analyticities or cusps near a topological phasetransition driven by changes of global topologies rather than by the formationof conventional (local) order. The particular phase transitions betweentopologically ordered superfluids and non-topological superfluids studied inthis article are always accompanied by the emergence or disappearance ofMajorana boundary modes, while at the critical point these boundary modes areliberated and evolve into the bulk. Proliferation of massless Majorana fermionsat the transition leaves unique thermodynamic signatures. For a very broadclass of isotropic superfluids or superconductors (with or without timereversal symmetry), the resultant transitions in $d$ spatial dimensions will beof $(d+1)$th order. The $(d+1)$th order derivatives of the grand potential areeither discontinuous in even dimensions or logarithmically divergent in odddimensions. Bulk quantities such as compressibility can develop a cusp alongwith the change of topology of the superfluids. We compute quantitatively thecusp structures and further investigate the effect of interactions induced bymassive fluctuation fields. These cusps are direct manifestations of Majoranafermions and their interactions, and can be potentially considered as analternative approach for detecting these puzzling particles.

Cusps near Topological Phase Transitions: Signatures of Majorana fermions and interactions with fluctuations