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

Abstract

The Wiedemann-Franz law relates the low temperature (T) limit of the ratio W ≡ κ/(σT) of the thermal (κ) and electrical (σ) conductivities of metals to the universal Lorenz number L0 = (π 2 /3)(kB/e) 2 . This remarkable relationship is independent of the strength of the interactions between the electrons, relates macroscopic transport properties to fundamental constants of nature, and depends only upon the Fermi statistics and charge of the elementary quasiparticle excitations of the metal. It has been experimentally verified to high precision in a wide range of metals [1], and realizes a sensitive macroscopic test of the quantum statistics of the charge carriers. It is interesting to note the value of the WiedemannFranz ratio in some other important strongly interacting quantum systems. In superconductors, which have low energy bosonic quasiparticle excitations, σ is infinite for a range of T > 0, while κ is finite in the presence of impurities [2], and so W = 0. At quantum phase transitions described by relativistic field theories, such as the superfluid-insulator transition in the Bose Hubbard model, the low energy excitations are strongly coupled and quasiparticles are not well defined; in such theories the conservation of the relativistic stress-energy tensor implies that κ is infinite, and so W = ∞ [3]. Li and Orignac [4] computed W in disordered Luttinger liquids, and found deviations from L0, and found a non-zero universal value for W at the metal-insulator transition for spinless fermions. The present paper will focus on the quantum phase transition between a superconductor and a metal (a SMT). We will consider quasi-one dimensional nanowires with a large number of transverse channels (so that the electronic localization length is much larger than the mean free path (l)) which can model numerous recent experiments [5, 6, 7, 8, 9, 10, 11, 12, 13]. We will describe universal deviations in the value of W from L0, which can serve as sensitive tests of the theory in future experiments. The mean-field theory for the SMT goes back to the early work [14] of Abrikosov and Gorkov (AG): in one of the earliest discussions of a quantum phase transition, they showed that a large enough concentration of magnetic impurities could induce a SMT at T = 0. It has since been shown that such a theory applies in a large variety of situations with ‘pair-breaking’ perturbations: anisotropic superconductors with non-magnetic impurities [15], lower-dimensional superconductors with magnetic fields oriented in a direction parallel to the Cooper pair motion [16], and s-wave superconductors with inhomogeneity in the strength of the attractive BCS interaction [17]. Indeed, it is expected that pair-breaking is present in any experimentally realizable SMT at T = 0: in the nanowire experiments, explicit evidence for pairbreaking magnetic moments on the wire surface was presented recently by Rogachev et al. [13].