  Recent detections of gravitational waves from mergers of neutron stars (NSs)
and black holes (BHs) in the low and high-end mass gap regimes pose a puzzle to
standard stellar and binary evolution theory. Mass-gap mergers may originate
from successive mergers in hierarchical systems such as quadruples. Here, we
consider repeated mergers of NSs and BHs in stellar 2+2 quadruple systems, in
which secular evolution can accelerate the merger of one of the inner binaries.
Subsequently, the merger remnant may interact with the companion binary,
yielding a second-generation merger. We model the initial stellar and binary
evolution of the inner binaries as isolated systems. In the case of successful
compact object formation, we subsequently follow the secular dynamical
evolution of the quadruple system. When a merger occurs, we take into account
merger recoil, and model subsequent evolution using direct N-body integration.
With different assumptions on the initial properties, we find that the majority
of first-generation mergers are not much affected by secular evolution, with
their observational properties mostly consistent with isolated binaries. A
small subset shows imprints of secular evolution through residual eccentricity
in the LIGO band, and retrograde spin-orbit orientations. Second-generation
mergers are ~10^7 times less common than first-generation mergers, and can be
strongly affected by scattering (i.e., three-body interactions) induced by the
first-generation merger. In particular, scattering can account for mergers
within the low-end mass gap, although not the high-end mass gap. Also, in a few
cases, scattering could explain highly eccentric LIGO sources and negative
effective spin parameters.
