  A minimal model for the hybrid superconductor-semiconductor nanowire Majorana
platform is developed that fully captures the effects of the low-energy
renormalization of the nanowire modes arising from the presence of the parent
superconductor. In this model, the parent superconductor is an active component
that participates explicitly in the low-energy physics, not just a passive
partner that only provides proximity-induced Cooper pairs for the nanowire.
This treatment on an equal footing of the superconductor and the semiconductor
has become necessary in view of recent experiments, which do not allow a
consistent interpretation based just on the bare semiconductor properties. The
general theory involves the evaluation of the exact semiconductor Green
function that includes a dynamical self-energy correction arising from the
tunnel-coupled superconductor. General conditions for the emergence of
topological superconductivity are worked out for single-band as well as
multi-band nanowires and detailed numerical results are given for both infinite
and finite wire cases. The topological quantum phase diagrams are provided
numerically and the Majorana bound states are obtained along with their
oscillatory energy splitting behaviors due to wavefunction overlap in finite
wires. Renormalization effects are shown to be both qualitatively and
quantitatively important in modifying the low-energy spectrum of the nanowire.
The results of the theory are found to be in good qualitative agreement with
Majorana nanowire experiments, leading to the conclusion that the
proximity-induced low-energy renormalization of the nanowire modes by the
parent superconductor is of fundamental importance in
superconductor-semiconductor hybrid structures. Implications of the general
theory for obtaining true zero energy topological Majorana modes are pointed
out.
