Absence of coherent peaks in a $Z_2$ fractionalized BCS superconducting state

We explore a $Z_2$ fractionalized Bardeen-Cooper-Schrieffer (BCS) superconducting state, which is a minimal extension of usual BCS framework. It is found that this state has similar thermal and transport properties, but its single-particle feature strongly deviates from the coherent quasiparticle behavior of the classic/conventional BCS superconducting state. The fingerprint of such $Z_2$ BCS state is the absence of the BCS coherent peaks and instead a kink in the local density of state occurs, which in principle could be probed by scanning tunneling microscopy or point-contact spectroscopy experiments. The corresponding exactly soluble models that realize the desirable $Z_2$ fractionalized BCS state is presented. In addition, we also study the extended $t$-$U$-$J$ model by using $Z_2$ slave-spin representation and find that the $Z_2$ BCS state may exist when the paring structure is fully gapped or has nodes. The prototypical wave-function of such a $Z_2$ BCS state is also proposed, which could be taken as trial wave-function in current numerical techniques. The present work may be helpful to further study the unconventional superconductivity and its relation to non-Fermi liquids.