Critical dynamics of thermal conductivity at the normal-superconducting transition

Some 15 years after the discovery of the cuprate superconductors, many tantalizing puzzles remain, particularly with regards to ground-state and possible quantum critical properties with varied doping. It has been difficult to disentangle consistent analyses faithful to experiment from the zoo of microscopic theories and models. In contrast, there have been notable successes in understanding the phenomenology of the finite temperature transition into the superconducting phase, where quantum fluctuations are unimportant and progress can be made employing classical LandauGinzburg approaches. In particular, at optimal doping in YBCO, the observed critical behavior in the electrical conductivity, specific heat, penetration depth, and other quantities appears to be consistent with theoretical expectations. Recent attention has focused on thermal transport experiments, which have shed light on the low-temperature transport of quasiparticles in the superconducting phase, and have revealed low-temperature violations of the WiedemannFranz law in the normal state of the electron-doped material suggestive of a non-Fermi liquid ground state. Here, we revisit the theory of thermal conductivity, focusing on the critical behavior near the finite temperature superconducting transition where progress is possible without the need for a microscopic quantum model. Older works within a BCS framework generalizing the Aslamazov-Larkin calculations to thermal conductivity, have predicted a diverging thermal conductivity upon cooling into the superconductor, reminiscent of the behavior of He at the l transition. This appears to be at odds with experiment in the cuprates, which typically show a finite and rather smooth thermal conductivity as one cools through Tc , with a large growth upon further cooling usually ascribed to quasiballistic-quasiparticle transport. Our study focuses on the three-dimensional disordered superconductor, most appropriate to optimally doped YBCO, which is the least two-dimensional cuprate. We follow the phenomenological hydrodynamic approach to critical dynamics pioneered by Hohenburg and Halperin. Indeed, one of the early successes of this dynamical scaling approach was the correct description of the diverging thermal conductivity near the l transition in He. Here, we modify this theory to account for impurities and long-ranged Coulomb forces appropriate to the superconductor. Our central conclusion is that rather than a divergent thermal conductivity as in