Superconductivity at a quantum critical point: A theoretical approach to the understanding of unconventional superconductors in strongly correlated systems.

Abstract

The complexity of strongly correlated electronic system is manifested by the interplay of multiple electronic orders, among which superconductivity is one of the most interesting phases. In experimentally observed phase diagrams for materials such as heavy fermion compounds, cuprates and iron based superconductors, superconductivity is close to other electronic orders such as ferromagnetism, anti-ferromagnetism and nematicity etc. This fact brings about interests of studying the role of a possible underlying quantum critical point(QCP) in determining the unusual properties of these materials. Here we consider an itinerant fermion system which is close to a QCP. Because of the closeness, the collective boson mode due to the order parameter fluctuations will couple to low energy fermions and mediate the fermion-fermion interaction. This effective interaction simultaneously gives rise to two competing fate for the fermions: On one hand it can lead to SC if the there is any pairing instability in at least one pairing channel. On the other hand, the same interaction also diminish fermion coherence and results in non Fermi liquid behavior. These two tendencies compete with each other, in a sense that SC gaps out low energy fermions and reduces the self energy , while non Fermi liquid tends to destroy fermion coherence and is detrimental to SC. In order to capture this story we adopt the approach that Eliashberg first used when he studied the electron-phonon coupling system, i.e. we approximate the fermion self energy by neglecting the vertex corrections, which is controllable when the vertex is parametrically smaller. We further assume the interaction depends only on frequency via a dynamical exponent $\gamma$, namely $V(\Omega_m)\propto 1/|\Omega_m|^\gamma$. Based on this model, we unveil many special properties of SC state on both imaginary and real frequency axis, including the ‘gap closing’ behavior observed in cuprates. As a unique feature of pairing at a QCP, we find there exists an infinite set of solutions to the gap equation, corresponding to different local minima in free energy. This set becomes a continuous one at a special case $\gamma=2$, which corresponds to phonon-mediated pairing interaction with a vanishingly small phonon frequency. We also studied the odd-frequency pairing state from this model, and find there is no zero bias peak in the quasiparticle density of states which was considered as an evidence of odd- frequency pairing. At last, in addition to mean field analysis, the superconducting phase fluctuation is also discussed.