  We propose that light-matter coupling can be used to realize synthetic
lattices. In particular, we consider a one-dimensional chain of exciton-photon
sites to create a comb lattice that exhibits a transition from a flat band to a
finite mass dispersion by tuning site-dependent light-matter coupling.
Moreover, in a non-Hermitian system with gain and loss, the flat band phase is
much more robust and the transition is accompanied by the appearance of
exceptional points in the complex energy spectrum. We demonstrate that by
engineering the light-matter coupling in the synthetic lattice, one can explore
various phases in the lasing regime. Our proposal paves the way for studying
non-Hermitian systems in higher dimensions.
