  We study the effect of Hund's splitting of repulsive interactions on
electronic phase transitions in the multiorbital topological crystalline
insulator Pb$_{1-x}$Sn$_{x}$Te, when the chemical potential is tuned to the
vicinity of low-lying Type-II Van Hove singularities. Nontrivial Berry phases
associated with the Bloch states impart momentum-dependence to electron
interactions in the relevant band. We use a multipatch parquet renormalization
group (RG) analysis for studying the competition of different electronic
phases, and find that if the dominant fixed-point interactions correspond to
antiparallel spin configurations, then a chiral $p$-wave
Fulde-Ferrell-Larkin-Ovchinnikov(FFLO) state is favored, otherwise, none of the
commonly encountered electronic instabilities occur within the one-loop parquet
RG approach.
