Exceptional points and spectral cusps from density-wave fluctuation

We report two types of singularities that arise from fluctuations during the formation of charge- or spin-density waves. The first is the exceptional point (EP), corresponding to a higher-order pole of the retarded Green's function. Such EPs lead to algebraic corrections in the decay of quasiparticle occupations and are observable through time-resolved angle-resolved photoemission spectroscopy (Tr-ARPES). The second is a spectral cusp, defined by the coalescence of three extrema in the real-frequency spectral function $A(\mathbf{k}, \omega)$. This cusp enforces the formation of Fermi arcs and induces a "threading" structure in the nearby band structure, both of which are directly observable in ARPES.