Spin-Hall Conductivity in Electron-Phonon Coupled Systems

We derive the ac spin-Hall conductivity $\sigma_{\rm sH}(\omega)$ of two-dimensional spin-orbit coupled systems interacting with dispersionless phonons of frequency $\omega_0$. For the linear Rashba model we show that the electron-phonon contribution to the spin-vertex corrections breaks the universality of $\sigma_{\rm sH}(\omega)$ at low-frequencies and provides a non-trivial renormalization of the interband resonance. On the contrary, in a generalized Rashba model for which the spin-vertex contributions are absent, the coupling to the phonons enters only through the self-energy, leaving the low frequency behavior of $\sigma_{\rm sH}(\omega)$ unaffected by the electron-phonon interaction.Comment: 4 pages, 3 figures, version as printe

Even-Odd and Super-Even Effects in the Attractive Hubbard Model

The canonical BCS wave function is tested for the attractive Hubbard model. Results are presented for one dimension, and are compared with the exact solutions by the Bethe ansatz and the results from the conventional grand canonical BCS approximation, for various chain lengths, electron densities, and coupling strengths. While the exact ground state energies are reproduced very well both by the canonical and grand canonical BCS approximations, the canonical method significantly improves the energy gaps for small systems and weak coupling. The ``parity'' effect due to the number of electrons being even or odd naturally emerges in our canonical results. Furthermore, we find a ``super-even'' effect: the energy gap oscillates as a function of even electron number, depending on whether the number of electrons is $4 m$ or $4 m + 2$ (m integer). Such oscillations as a function of electron number should be observable with tunneling measurements in ultrasmall metallic grains.Comment: 20 pages, 9 figure

Optical conductivity in non-equilibrium d-wave superconductors

We consider the optical conductivity of a d-wave BCS superconductor in the presence of a non-equilibrium distribution of excess quasiparticles. Two different simplified models used in the past for the s-wave case are considered and results compared. In the $T^\ast$-model of Parker the excess quasiparticles are assumed to be in a thermal distribution at some temperature $T^\ast$ larger than the equilibrium sample temperature. In the $\mu^\ast$- model of Owen and Scalapino a chemical potential is introduced to accommodate the excess quasiparticles. Some of the results obtained are specific to the model, most are qualitatively similar in both.Comment: 11 pages, 6 figures this manuscript has been accepted for publication in abbreviated form by Physical Review