  Many features of charge-4e superconductors remain unknown because even the
"mean-field Hamiltonian" describing them is an interacting model. Here we
introduce an interacting model to describe a charge-4e superconductor (SC) deep
in the superconducting phase, and explore its properties using quantum Monte
Carlo (QMC) simulations. The QMC is sign-problem-free, but only when a Majorana
representation is employed. As a function of the chemical potential we observe
two sharply-distinct behaviors: a "strong" quarteting phase in which charge-4e
quartets are tightly bound (like molecules) so that charge-2e pairing does not
occur even in the temperature $T \to 0$ limit, and a "weak" quarteting phase in
which a further transition to a charge-2e superconducting phase occurs at a
lower critical temperature. Analogous issues arise in a putative Z4 spin-liquid
with a pseudo-Fermi surface and in characterizing charge-2e SCs with "odd
frequency pairing" (or equivalently with a "composite order parameter") and
metallic spin-nematics. Under certain circumstances, we also identified a
stable T=0 charge-4e SC phase with gapless nodal quasiparticles. We further
discuss possible relevance of our results to various experimental observations
in 1/8-doped LBCO.
