  Motivated by the recently developed classical spin model for
Migdal-Eliashberg theory, we develop new numerical and analytical methods based
on this spin-chain representation and apply these methods to the
Bogoliuov-Tomachov-Morel-Anderson pairing potential, which incorporates the
phonon-mediated attraction and Coulomb repulsion. We show that the Monte Carlo
method with heat bath updates can efficiently obtain the gap functions even for
the situations challenging for the iterative solvers, suggesting an
unprecedented robust approach for solving the full nonlinear Migdal-Eliashberg
theory. Moreover, we derive the renormalization of all the couplings by tracing
out the high-frequency spins in the partition function. The derived analytical
renormalization equations produce the well-known $\mu^*$ effect for the
Bogoliuov-Tomachov-Morel-Anderson pairing potential and can be generalized to
other superconductivity problems. We further point out that several interesting
features (e.g., sign changing in the frequency-dependent gap function) can be
intuitively understood using the classical spin-chain representation for
Migdal-Eliasherg theory. Our results show the advantage of using the spin-chain
representation for solving Migdal-Eliashberg theory and provide new ways for
tackling general superconductivity problems.
