Superconductors with Magnetic Impurities: Instantons and Sub-gap States

When subject to a weak magnetic impurity potential, the order parameter and quasi-particle energy gap of a bulk singlet superconductor are suppressed. According to the conventional mean-field theory of Abrikosov and Gor'kov, the integrity of the energy gap is maintained up to a critical concentration of magnetic impurities. In this paper, a field theoretic approach is developed to critically analyze the validity of the mean field theory. Using the supersymmetry technique we find a spatially homogeneous saddle-point that reproduces the Abrikosov-Gor'kov theory, and identify instanton contributions to the density of states that render the quasi-particle energy gap soft at any non-zero magnetic impurity concentration. The sub-gap states are associated with supersymmetry broken field configurations of the action. An analysis of fluctuations around these configurations shows how the underlying supersymmetry of the action is restored by zero modes. An estimate of the density of states is given for all dimensionalities. To illustrate the universality of the present scheme we apply the same method to study `gap fluctuations' in a normal quantum dot coupled to a superconducting terminal. Using the same instanton approach, we recover the universal result recently proposed by Vavilov et al. Finally, we emphasize the universality of the present scheme for the description of gap fluctuations in d-dimensional superconducting/normal structures.Comment: 18 pages, 9 eps figure

Cosmological Constraints on Decaying Dark Matter

We present a complete analysis of the cosmological constraints on decaying dark matter. Previous analyses have used the cosmic microwave background and Type Ia supernova. We have updated them with the latest data as well as extended the analysis with the inclusion of Lyman-$\alpha$ forest, large scale structure and weak lensing observations. Astrophysical constraints are not considered in the present paper. The bounds on the lifetime of decaying dark matter are dominated by either the late-time integrated Sachs-Wolfe effect for the scenario with weak reionization, or CMB polarization observations when there is significant reionization. For the respective scenarios, the lifetimes for decaying dark matter are $\Gamma^{-1} \gtrsim 100$ Gyr and $(f \Gamma) ^{-1} \gtrsim 5.3 \times 10^8$ Gyr (at 95.4% confidence level), where the phenomenological parameter $f$ is the fraction of the decay energy deposited in baryonic gas. This allows us to constrain particle physics models with dark matter candidates through investigation of dark matter decays into Standard Model particles via effective operators. For decaying dark matter of $\sim 100$ GeV mass, we found that the size of the coupling constant in the effective dimension-4 operators responsible for dark matter decay has to generically be $\lesssim 10^{-22}$. We have also explored the implications of our analysis for representative models in theories of gauge-mediated supersymmetry breaking, minimal supergravity and little Higgs.Comment: 29 pages, 6 figures. Added references and corrected typos as well as grammatical oversight