Topological Phase Transition between $s^{+-}$ and $s^{++}$ Superconducting Phases from Competing Electron--Electron and Electron--Phonon Interactions

We discover a topological phase transition between conventional $s^{+-}$ and $s^{++}$ superconducting phases by tuning the ratio of electron--electron and electron--phonon coupling constants in an FeAs-type two-band structure. Proving the existence of this unexpected quantum criticality within the mean-field theory, we propose that the quantum critical point be identified with a critical spin liquid state of an "extended" Dirac spectrum, where critical superconducting fluctuations cause screening of charge degrees of freedom for electronic excitations, which allows spinon excitations to carry only the spin quantum number 1/2. The emergence of the critical spin liquid state at the $s^{+-}-s^{++}$ superconducting quantum critical point leads us to predict a metal--insulator--metal crossover behavior in electrical resistivity above the superconducting transition temperatures as the ratio of the electron--electron and electron--phonon coupling constants is increased. In addition, we uncover that the competition between electron--electron repulsion and electron--phonon attraction gives rise to a huge enhancement of the superconducting transition temperature near the quantum critical point which is several hundreds percent larger than that of the case when only one of the two is taken into account. Our renormalization group analysis claims that this mechanism for the enhancement of the critical temperature is not limited to superconductivity but can be applied to various Fermi surface instabilities, proposing an underlying universal structure, which turns out to be essentially identical to that of a recent study [Phys. Rev. Lett. 108 (2012) 046601] on the enhancement of the Kondo temperature in the presence of Rashba spin--orbit interaction. We speculate that the existence of this possible "deconfined" quantum criticality can be verified not only theoretically but also experimentally, particularly, in Li 2 (Pd 1-x Pt x ) 3 B superconductors, varying $x$ from 0 to 1.