Universal thermal and electrical transport near the superconductor-metal quantum phase transition in nanowires

We describe the thermal $\left(\kappa \right)$ and electrical $\left(\sigma \right)$ conductivities of quasi-one-dimensional wires, across a quantum phase transition from a superconductor to a metal induced by pair-breaking perturbations. Fluctuation corrections to BCS theory motivate a field theory for quantum criticality. We describe deviations in the Wiedemann-Franz ratio $\kappa ∕\sigma T$ (where $T$ is the temperature) from the Lorenz number $({\pi }^2∕3){(k_B∕e)}^2$, which can act as sensitive tests of the theory. We also describe the crossovers out of the quantum-critical region into the metallic and superconducting phases.

Figure 1

(Color online) Crossover phase diagram of the superconductor-metal transition in a quasi-one-dimensional superconductor. The “Metal” is described by the perturbative theory of Ref. 2. The “Quantum-critical” region is described by $\mathcal{S}$ and realizes our result for $W$ in Eq. (1) for temperatures above $T_{\mathrm{dis}}$ where the effects of disorder may be neglected. The Mooij-Schön mode is present everywhere, but strongly couples to superconducting fluctuations only in the “Fluctuating superconductor” regime, where it is described by Eq. (11); note that $\mathcal{S}$ does not apply here. The dashed lines are crossover boundaries which by Eq. (4) occur at $T\sim {\mid \alpha -{\alpha }_c\mid }^{z\nu }$.