  Yang and Lee investigated phase transitions in terms of zeros of partition
functions, namely, Yang-Lee zeros [Phys. Rev. 87, 404 (1952); Phys. Rev. 87,
410 (1952)]. We show that the essential singularity in the superconducting gap
is directly related to the number of roots of the partition function of a BCS
superconductor. Those zeros are found to be distributed on a semicircle in the
complex plane of the interaction strength due to the Fermi-surface instability.
A renormalization-group analysis shows that the semicircle theorem holds for a
generic quantum many-body system with a marginal coupling, in sharp contrast
with the Lee-Yang circle theorem for the Ising spin system. This indicates that
the geometry of Yang-Lee zeros is directly connected to the Fermi-surface
instability. Furthermore, we unveil the nonunitary criticality in BCS
superconductivity that emerges at each individual Yang-Lee zero due to
exceptional points and presents a universality class distinct from that of the
conventional Yang-Lee edge singularity.
