  Superconductivity is an emergent phenomena in the sense that the energy scale
associated with Cooper pairing is generically much lower than the typical
kinetic energy of electrons. Addressing the mechanism of Cooper pairing amounts
to determine the effective interaction that operates at low energies. Deriving
such an interaction from a bottom-up approach has not been possible for any
superconductor, especially strongly correlated ones. Top-down approaches, where
one assumes an effective interaction, is plagued with the difficulty of
extracting the implied electronic instabilities without uncontrolled
approximations. These facts severely hinder our ability to determine the
pairing mechanism for high temperature superconductors. Here we perform
large-scale sign-problem-free quantum Monte-Carlo simulations on an effective
theory, featured with antiferromagnetic and nematic fluctuations, to study the
intertwined antiferromagnetic, superconducting, and charge density wave
instabilities of the cuprates. Our results suggest the inclusion of nematic
fluctuations is essential in order to produce the observed type of charge
density wave ordering. Interestingly we find that the d-wave Cooper pairing is
enhanced by nematic fluctuations.
