  Using density functional plus dynamical mean-field theory method (DFT+DMFT)
with full self-consistency over the charge density, we study the effect of
electronic correlations on the electronic structure, magnetic properties,
orbital-dependent band renormalizations, and Fermi surface of the tetragonal
phase of bulk FeS. We perform a direct structural optimization of the $P_4/nmm$
crystal structure of paramagnetic FeS, with respect to the lattice constant $a$
and the internal coordinate $z_\mathrm{S}$ of atom S. Our results show an
anomalous sensitivity of the electronic structure and magnetic properties of
FeS to fine details of its crystals structure. Upon expansion of the lattice
volume, we observe a remarkable change of the electronic structure of FeS which
is associated with a complete reconstruction of the Fermi surface topology
(Lifshitz transition). This behavior is ascribed to a correlation-induced shift
of the Van Hove singularity associated with the Fe $t_2$ orbitals at the $M$
point across the Fermi level. The Lifshitz phase transition is accompanied by a
significant growth of local magnetic moments and emergence of strong
orbital-selective correlations. It is seen as a pronounced anomaly (`kink') in
the total energies upon expansion of the lattice, associated with a remarkable
enhancement of compressibility. This behavior is accompanied by an
orbital-dependent formation of local moments, a crossover from itinerant to
localized orbital-selective moment behavior of the Fe $3d$ electrons. While
exhibiting weak effective mass enhancement of the Fe $3d$ states $m^*/m \sim
1.3-1.4$, correlation effects reveal a strong impact on a position of the Van
Hove singularity at the $M$ point, implying a complex interplay between
electronic correlations and band structure effects in FeS.

Strongly Correlated Electrons