Monte Carlo solver and renormalization of Migdal-Eliashberg spin chain

Motivated by the recently developed classical spin model for the Migdal-Eliashberg theory, we develop numerical and analytical methods based on this spin-chain representation and apply these methods to the Bogoliubov-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 Bogoliubov-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.