Interplay between superconductivity and non-Fermi liquid behavior at a quantum critical point in a metal. III. The γ model and its phase diagram across γ =1
In this paper we continue our analysis of the interplay between the pairing and the non-Fermi liquid behavior in a metal for a set of quantum-critical models with an effective dynamical electron-electron interaction V (Ωm) ∝1 /|Ωm| γ (the γ model). We analyze both the original model and its extension, in which we introduce an extra parameter N to account for nonequal interactions in the particle-hole and particle-particle channel. In two previous papers [A. Abanov and A. V. Chubukov, Phys. Rev. B 102, 024524 (2020), 10.1103/PhysRevB.102.024524 and Y. Wu et al. Phys. Rev. B 102, 024525 (2020), 10.1103/PhysRevB.102.024525] we considered the case 0 <γ <1 and argued that (i) at T =0 , there exists an infinite discrete set of topologically different gap functions Δn(ωm) , all with the same spatial symmetry, and (ii) each Δn evolves with temperature and terminates at a particular Tp ,n. In this paper we analyze how the system behavior changes between γ <1 and γ >1 , both at T =0 and a finite T . The limit γ →1 is singular due to infrared divergence of ∫d ωmV (Ωm) , and the system behavior is highly sensitive to how this limit is taken. We show that for N =1 , the divergencies in the gap equation cancel out, and Δn(ωm) gradually evolve through γ =1 both at T =0 and a finite T . For N ≠1 , divergent terms do not cancel, and a qualitatively new behavior emerges for γ >1 . Namely, the form of Δn(ωm) changes qualitatively, and the spectrum of condensation energies Ec ,n becomes continuous at T =0 . We introduce different extension of the model, which is free from singularities for γ >1 .
Physical Review B
- Pub Date:
- September 2020
- Condensed Matter - Superconductivity;
- Condensed Matter - Strongly Correlated Electrons
- 43 pages, 25 figures, Paper III in the series on the gamma-model. Papers I and II are arXiv:2004.13220 and arXiv:2006.02968 respectively