Nodeless superconducting gap induced by odd parity pair density wave in underdoped cuprates

We present a theoretical model to show that the transition from an antiferromagnetic (AFM) phase to the $d_{x^2-y^2}$-wave superconductivity occurs through a robust companion of a finite momentum pairing between $({\bf k},\uparrow)$ and $(-{\bf k}-{\bf Q},\downarrow)$ electrons, namely pair-density wave (PDW) state, where ${\bf Q}=(\pi,\pi)$ is the AFM wavevector. Interestingly, the spatial structure of the PDW is constrained to be a $p+ip$-wave symmetry, which follows $p_{\bf k}=p_{-{\bf k}-{\bf Q}}$ under fermions exchange in cuprates, dictating a spin-singlet pairing. Furthermore, the PDW state produces a fully gapped quasiparticle spectrum which explains recent observations of the fully gapped quasiparticle structure of lightly doped cuprates in the hole doping side. We study the stability of all three phases within the self-consistent mean-field theory. Finally, we calculate the superfluid density to propose its exponential temperature dependence as a test to the SC origin of this fully gapped state, while the phase modulation of the PDW state can be visualized by scanning probes.