In the first part, I will discuss superconductivity in doped SrTiO₃ (STO), where the pairing
mechanism remains unresolved. Several experiments have observed an enhancement of Tc
near a ferroelectric quantum critical point (QCP) associated with the softening of a
transverse optical (TO) phonon mode. This suggests that critical ferroelectric fluctuations
play an important role in pairing in STO. Using strong-coupling Eliashberg theory, we have
shown that a linear Rashba-type coupling between electrons and the soft TO mode captures
the main features of the experimentally observed superconducting phase diagram, while an
additional nonlinear (quadratic) coupling is needed for agreement with its finer features.
In the second part, I will turn to the application of complex-path quantum Monte Carlo (QMC)
to superconducting systems with repulsive interactions. Here, repulsion generates complex
weights, leading to strong phase cancellations and a severe sign problem in conventional
QMC. To address this challenge, we deform the path integral contour into the complex plane
using holomorphic flow, which preserves the partition function while guiding the sampling
manifold toward steepest-descent regions and reducing the sign problem. As a first step, I
have applied this approach to a simplified Morel–Anderson interaction, neglecting spatial
and dynamical fluctuations. I will show that this method dramatically reduces the sign
problem and enables calculations at much lower temperatures. This work provides a
foundation for extending this method to more realistic systems with both spatial and temporal
fluctuations.