Multiple solutions of pairing gap equation in quantum critical metals.

We use Eliashberg theory to analyze superconductivity for a class of quantum-critical models of itinerant fermions interacting with collective massless bosons, with varying scaling dimension γ of a boson (the γ model). A conventional wisdom holds that there is a single T_c for the pairing in a given symmetry channel. We find in this study that at the critical point the situation is different: the linearized gap equation has a cascade of solutions at T=T_c⁽ⁿ⁾ . These solutions have the same spatial symmetry, but they are topologically distinct by the number of sign changes as functions of Matsubara frequency. The transition temperatures T_c⁽ⁿ⁾ decrease exponentially with increasing number of nodes n, and the largest T_c⁽⁰⁾ has no nodes. We further show that below a given T_c⁽ⁿ⁾ , the corresponding solution Δ(ω_n) of the linearized gap equation grows in magnitude, but maintains the number of sign changes, which in this respect acts as a topological invariant. We discuss how these oscillating solutions evolve with increasing scaling dimension γ, and how they contribute to destruction of long range superconducting order.