Hydrogen-based compounds under ultrahigh pressure, such as the polyhydrides ${\mathrm{H}}_3\mathrm{S}$ and ${\mathrm{LaH}}_{10}$, superconduct through the conventional electron-phonon coupling mechanism to attain the record critical temperatures known to date. Here we exploit the intrinsic advantages of hydrogen to strongly enhance phonon-mediated superconductivity in a completely different system, namely, a two-dimensional material with hydrogen adatoms. We find that van Hove singularities in the electronic structure, originating from atomiclike hydrogen states, lead to a strong increase of the electronic density of states at the Fermi level, and thus of the electron-phonon coupling. Additionally, the emergence of high-frequency hydrogen-related phonon modes in this system boosts the electron-phonon coupling further. As a concrete example, we demonstrate the effect of hydrogen adatoms on the superconducting properties of monolayer ${\mathrm{MgB}}_2$, by solving the fully anisotropic Eliashberg equations, in conjunction with a first-principles description of the electronic and vibrational states, and their coupling. We show that hydrogenation leads to a high critical temperature of 67 K, which can be boosted to over 100 K by biaxial tensile strain.

Starting from first principles, we show the formation and evolution of superconducting gaps in ${\mathrm{MgB}}_2$ at its ultrathin limit. Atomically thin ${\mathrm{MgB}}_2$ is distinctly different from bulk ${\mathrm{MgB}}_2$ in that surface states become comparable in electronic density to the bulklike $\sigma$ and $\pi$ bands. Combining the ab initio electron-phonon coupling with the anisotropic Eliashberg equations, we show that monolayer ${\mathrm{MgB}}_2$ develops three distinct superconducting gaps, on completely separate parts of the Fermi surface due to the emergent surface contribution. These gaps hybridize nontrivially with every extra monolayer added to the film owing to the opening of additional coupling channels. Furthermore, we reveal that the three-gap superconductivity in monolayer ${\mathrm{MgB}}_2$ is robust over the entire temperature range that stretches up to a considerably high critical temperature of 20 K. The latter can be boosted to $>50\mathrm{K}$ under biaxial tensile strain of $\sim 4\%$, which is an enhancement that is stronger than in any other graphene-related superconductor known to date.

We develop the anisotropic Eliashberg framework for superconductivity in the presence of an applied magnetic field. Using as input the ab initio calculated electron and phonon band structures and electron-phonon coupling, we solve self-consistently the anisotropic Eliashberg equations for the archetypal superconductor ${\mathrm{MgB}}_2$. We find two self-consistent solutions, time-even two-band superconductivity, as well as unconventional time-odd $s$-wave spin triplet two-band superconductivity emerging with applied field. We provide the full momentum, frequency, and spin-resolved dependence and magnetic field-temperature phase diagrams of the time-even and time-odd superconducting pair amplitudes and predict fingerprints of this novel odd-frequency state in tunneling experiments.

CeCoIn5 is an anomalous superconductor which exhibits a high-magnetic-field phase that consists of a modulated magnetic coupling together with persistent superconducting order. Here we use a generic microscopic model to argue that this state is a pattern of coexisting condensates: a d-wave singlet superconducting (SC) state, a staggered $\pi${}-triplet SC state, and a spin density wave (SDW). Our microscopic picture allows a calculation of the phase diagram, and physical consequences including NMR. We interpret the appearance of the SDW order in the Q~phase as being induced by odd-triplet pairing.

We examine the impact of interfacial phonons on the superconducting state of $\mathrm{FeSe}/{\mathrm{SrTiO}}_3$ developing a material's specific multiband, full bandwidth, and anisotropic Eliashberg theory for this system. Our self-consistent calculations highlight the importance of the interfacial electron-phonon interaction, which is hidden behind the seemingly weak-coupling constant ${\lambda }_m=0.4$, in mediating the high $T_c$, and explain other puzzling experimental observations, such as the $s$-wave symmetry and replica bands. We discover that the formation of replica bands has a $T_c$ decreasing effect that is nevertheless compensated by deep Fermi-sea Cooper pairing which has a $T_c$ enhancing effect. We predict a strong-coupling dip-hump signature in the tunneling spectra due to the interfacial coupling.