Low-temperature quasiparticle transport in a $d$-wave superconductor with coexisting charge order

In light of the evidence that charge order coexists with $d$-wave superconductivity in the underdoped cuprate superconductors, we investigate the manner in which such charge order will influence the quasiparticle excitations of the system and, in particular, the low-temperature transport of heat by those quasiparticles. We consider a d-wave superconductor in which the superconductivity coexists with charge density wave order of wave vector $\left(\pi /a,0\right)$. While the nodes of the quasiparticle energy spectrum survive the onset of charge order, there exists a critical value of the charge density wave order parameter beyond which the quasiparticle spectrum becomes fully gapped. We perform a linear response Kubo formula calculation of thermal conductivity in the low temperature (universal) limit. Results reveal the dependence of thermal transport on increasing charge order up to the critical value at which the quasiparticle spectrum becomes fully gapped and thermal conductivity vanishes. In addition to numerical results, closed-form expressions are obtained in the clean limit for the special case of isotropic Dirac nodes. Signatures of the influence of charge order on low-temperature thermal transport are identified.

Figure 1

Charge order of wave vector $\mathbf{Q}=\left(\pi /a,0\right)$ doubles the unit cell and thereby halves the Brillouin zone. With increasing charge density wave order parameter, $\psi$, the nodes of the energy spectrum, and their images in the second reduced Brillouin zone (shaded), approach the reduced Brillouin zone edges (dotted), colliding for $\psi ={\psi }_c$, beyond which the spectrum is fully gapped.

Figure 2

Local coordinates, $k_1$ and $k_2$, defined about each of the four nodal collision points, ${\mathbf{k}}_c=\left(\pm \pi /2a,\pm \pi /2a\right)$. The $k_1$ axes, perpendicular to the Fermi surface, define the direction of increasing electron dispersion, ${ϵ}_k$. The $k_2$ axes, parallel to the Fermi surface, define the direction of increasing superconducting order parameter, ${\Delta }_k$.

Figure 3

Feynman diagram depicting the bare bubble thermal current-current correlation function, ${\stackrel{⃡}{\Pi }}_{\kappa }\left(i\Omega \right)$. The thermal current operator sits on each vertex and each propagator denotes a Green’s function dressed with disorder self-energy.

Figure 4

Calculated zero-temperature thermal conductivity tensor in the clean $\left({\Gamma }_0\to 0\right)$, isotropic $\left(v_F=v_{\Delta }\right)$ limit. We plot ${\kappa }_0^{xx}$ and ${\kappa }_0^{yy}$ as functions of the charge density wave order parameter, $\psi$, from the closed-form expressions in Eqs. (80, 81). As $\psi$ approaches ${\psi }_c$, the value beyond which the quasiparticle spectrum becomes gapped, ${\kappa }_0^{xx}$ vanishes continuously while ${\kappa }_0^{yy}$ diverges before dropping to zero.

Figure 5

Disorder dependence of calculated zero-temperature thermal conductivity tensor. We plot ${\kappa }_0^{xx}$ (upper panel) and ${\kappa }_0^{yy}$ (lower panel) as functions of charge density wave order parameter, $\psi$, for several values of the disorder parameter, ${\Gamma }_0$. In all cases, $\alpha =v_F/v_{\Delta }=1$. Included for comparison is the clean limit result (solid lines). Note that disorder smoothes the transition to zero thermal conductivity that results from the gapping of the energy spectrum at $\psi ={\psi }_c$. The divergence of ${\kappa }_0^{yy}$ seen in the clean case is replaced by a peak that is diminished and broadened with increasing disorder.

Figure 6

Nodal anisotropy dependence of calculated zero-temperature thermal conductivity tensor. For fixed disorder $\left({\Gamma }_0=0.05{\psi }_c\right)$, we plot ${\kappa }_0^{xx}$ (upper panel) and ${\kappa }_0^{yy}$ (lower panel) as functions of charge density wave order parameter, $\psi$, for several values of $\alpha =v_F/v_{\Delta }$. Lines connecting the data points are guides to the eye. Note that the nodal transition at $\psi ={\psi }_c$ is smoothed out by increasing velocity anisotropy in a manner similar to, but distinct from, the effect of disorder.