The ??-model: density of states of low-energy fermionic excitations

We study the competition between non-Fermi liquid and superconductivity near a quantum critical point of a metal, where  critical bosonic fluctuations mediate the interaction between electrons. When the Fermi energy is much larger than the typical bosonic energy, the low-energy properties of the system are effectively described by effective Eliashberg theory with singular bosonic propagator 1/|??|^?? (the  ??-model). This talk is based on our series of studies on this model and particularly focuses on the density of states (DOS) of the low-energy fermionic excitations. The normal state is a non-Fermi liquid state due to the singular self-energy at low frequencies and the ground state is a superconductor. At zero temperature, the DOS vanishes below the pairing gap and forms a continuum above. For small ??, the behavior of the DOS is qualitatively the same as the a BCS superconductor. As ?? increases, more spectral weight transfers to the lower-edge of the continuum. We discuss in detail how the DOS evolves with ??. At ??=2, the continuum collapses into a discrete set of ??-peaks, corresponding to the formation of polaronic bound states. For 2<??<3, the DOS again has a form of a gapped continuum, but contains a level with macroscopic degeneracy at the edge of the continuum.