Gordian knot in the pseudo‐gap phase of hole‐underdoped cuprates

We consider the coexistent d‐density wave (DDW) order, at the anti‐ferromagnetic (AF) wave vector Q  = ( π , π ), representing the pseudo‐gap (PG) state, and d‐wave superconductivity (DSC), driven by an assumed attractive interaction, for the two‐dimensional (2D) fermion system on a square lattice starting with a mean‐field Hamiltonian involving the singlet DDW and the DSC pairings. The two energy gaps, corresponding to PG and DSC, in the excitation spectrum do not merge into one ‘quadrature’ gap if the normal state dispersion is imperfectly nested. With the pinning of the van Hove singularities (vHSs) of the dispersion close to the chemical potential (µ), we show that the PG and DSC are representing two competing orders as the former brings about a depletion of the spectral weight (a contour plot of the spectral weight density for 60 K at the doping level ∼10% is shown in the abstract figure) available for pairing in the anti‐nodal region of momentum space in the coexistent state. The calculation of quasi‐particle thermal conductivity α xx in the DDW and normal phases via the Boltzmann equation in the relaxation‐time approximation shows that there is a mild discontinuity in α xx at the PG transition temperature T *.Contour plot of the spectral weight density on the first Brillouin zone for the coexisting pseudo‐gap (PG) and superconducting (SC) phases at 60 K. The gap amplitudes are $\Delta _{{\rm PG}}^{{\rm (0)}} /t_{1} $  ∼ 0.024 and $\Delta _{{\rm SC}}^{{\rm (0)}} /t_{1} $  ∼ 0.01, where t 1 corresponds to first‐neighbour hopping.