  In recent years, reduced basis methods (RBMs) have been adapted to the
many-body eigenvalue problem and they have been used, largely in nuclear
physics, as fast emulators able to bypass expensive direct computations while
still providing highly accurate results. This work is meant to show that the
RBM is an efficient and accurate emulator for the strong correlations induced
by the pairing interaction in a variety of finite systems like ultrasmall
superconducting grains, interacting topological superfluids and mesoscopic
hybrid superconductor-semiconductor devices, all of which require an expensive,
beyond-mean-field, particle-number conserving description. These systems are
modelled by the number-conserving Richardson pairing Hamiltonian and its
appropriate generalizations. Their ground state is solved for exactly using the
Density Matrix Renormalization Group. The reduced basis is assembled
iteratively from a small number of exact ground state vectors, well-chosen from
across the relevant parameter space using a fast estimate of the emulation
error and a greedy local optimization algorithm. The reduced basis emulation is
found to accurately describe the weak-to-strong pairing cross-over in small
grains, the third-order topological phase transition of the interacting
Richardson-Kitaev chain, and the complex charge stability diagram of a hybrid
quantum dot - superconductor device. RBMs are thus confirmed to be cheap and
accurate emulators for the widely encountered superconducting phenomena.
Capable of providing orders of magnitude computational speed-up with respect to
approaches based only on traditional many-body solvers, they open new
possibilities in building and solving models of interacting many-body systems
and in better interfacing them with experimental design and data analysis.
