  We demonstrate the existence of exceptional points of degeneracy (EPD) of
periodic eigenstates in non-Hermitian coupled chains of dipolar scatterers.
Guided modes supported by these structures can exhibit an EPD in their
dispersion diagram at which two or more Bloch eigenstates coalesce, in both
their eigenvectors and eigenvalues. We show a second-order modal EPD associated
with the parity-time ($\cal{PT}$) symmetry condition, at which each particle
pair in the double chain exhibits balanced gain and loss. Furthermore, we also
demonstrate a fourth-order EPD occurring at the band edge. Such degeneracy
condition was previously referred to as a degenerate band edge in lossless
anisotropic photonic crystals. Here, we rigorously show it under the occurrence
of gain and loss balance for a discrete guiding system. We identify a more
general regime of gain and loss balance showing that $\cal{PT}$-symmetry is not
necessary to realize EPDs. Furthermore, we investigate the degree of detuning
of the EPD when the geometrical symmetry or balanced condition is broken. These
findings open unprecedented avenues toward superior light localization and
transport with application to high-Q resonators utilized in sensors, filters,
low-threshold switching and lasing.
