Low temperature thermal conductivity in a d-wave superconductor with coexisting charge order: Effect of self-consistent disorder and vertex corrections

Given the experimental evidence of charge order in the underdoped cuprate superconductors, we consider the effect of coexisting charge order on low-temperature thermal transport in a d-wave superconductor. Using a phenomenological Hamiltonian that describes a two-dimensional system in the presence of a Q=(\pi,0) charge density wave and d-wave superconducting order, and including the effects of weak impurity scattering, we compute the self-energy of the quasiparticles within the self-consistent Born approximation, and calculate the zero-temperature thermal conductivity using linear response formalism. We find that vertex corrections within the ladder approximation do not significantly modify the bare-bubble result that was previously calculated. However, self-consistent treatment of the disorder does modify the charge-order-dependence of the thermal conductivity tensor, in that the magnitude of charge order required for the system to become effectively gapped is renormalized, generally to a smaller value.