Flux-Periodicity Crossover from hc/e in Normal Metallic to hc/2e in Superconducting Loops

The periodic response of a metallic or a superconducting ring to an external magnetic flux is one of the most evident manifestations of quantum mechanics. It is generally understood that the oscillation period hc/2e in the superconducting state is half the period hc/e in the metallic state, because the supercurrent is carried by Cooper pairs with a charge 2e. On the basis of the Bardeen-Cooper-Schrieffer theory we discuss, in which cases this simple interpretation is valid and when a more careful analysis is needed. In fact, the knowledge of the oscillation period of the current in the ring provides information on the electron interactions. In particular, we analyze the crossover from the hc/e periodic normal current to the hc/2e periodic supercurrent upon turning on a pairing interaction in a metal ring. Further, we elaborate on the periodicity crossover when cooling a metallic loop through the superconducting transition temperature Tc.Comment: To be bublished in "Superconductors", InTech (Rijeka), 2012 (ISBN 979-953-307-798-6

Superconductivity with Finite-Momentum Pairing in Zero Magnetic Field

In the BCS theory of superconductivity, one assumes that all Cooper pairs have the same center of mass momentum. This is indeed enforced by self consistency, if the pairing interaction is momentum independent. Here, we show that for an attractive nearest neighbor interaction, this is different. In this case, stable solutions with pairs with momenta q and -q coexist and, for a sufficiently strong interaction, one of these states becomes the groundstate of the superconductor. This finite-momentum pairing state is accompanied by a charge order with wave vector 2q. For a weak pairing interaction, the groundstate is a d-wave superconductor

Fractional Flux Quantization in Loops of Unconventional Superconductors

The magnetic flux threading a conventional superconducting ring is typically quantized in units of $\Phi_0=hc/2e$. The factor 2 in the denominator of $\Phi_0$ originates from the existence of two different types of pairing states with minima of the free energy at even and odd multiples of $\Phi_0$. Here we show that spatially modulated pairing states exist with energy minima at fractional flux values, in particular at multiples of $\Phi_0/2$. In such states condensates with different center-of-mass momenta of the Cooper pairs coexist. The proposed mechanism for fractional flux quantization is discussed in the context of cuprate superconductors, where $hc/4e$ flux periodicities as well as uniaxially modulated superconducting states were observed.Comment: 5 pages, 3 figure

Supercurrent as a Probe for Topological Superconductivity in Magnetic Adatom Chains

A magnetic adatom chain, proximity coupled to a conventional superconductor with spin-orbit coupling, exhibits locally an odd-parity, spin-triplet pairing amplitude. We show that the singlet-triplet junction, thus formed, leads to a net spin accumulation in the near vicinity of the chain. The accumulated spins are polarized along the direction of the local $\mathbf{d}$-vector for triplet pairing and generate an enhanced persistent current flowing around the chain. The spin polarization and the "supercurrent" reverse their directions beyond a critical exchange coupling strength at which the singlet superconducting order changes its sign on the chain. The current is strongly enhanced in the topological superconducting regime where Majorana bound states appear at the chain ends. The current and the spin profile offer alternative routes to characterize the topological superconducting state in adatom chains and islands.Comment: 5 pages, 3 figures, 5 pages of supplemental material

Momentum-Space Spin Texture in a Topological Superconductor

A conventional superconductor with spin-orbit coupling turns into a topological superconductor beyond a critical strength of the Zeeman energy. The spin-expectation values $\mathbf{S}(\mathbf{k})$ in momentum space trace this transition via a characteristic change in the topological character of the spin texture within the Brillouin zone. At the transition the skyrmion counting number switches from 0 to 1/2 identifying the topological superconductor via its meron-like spin texture. The change in the skyrmion counting number is crucially controlled by singular points of the map $\mathbf{S}(\mathbf{k})/|\mathbf{S}(\mathbf{k})|$ from the Brillouin zone, i.e. a torus, to the unit sphere. The complexity of this spin-map is discussed at zero temperature as well as for the extension to finite temperatures.Comment: 16 pages, 9 figure

All scale-free networks are sparse

We study the realizability of scale free-networks with a given degree sequence, showing that the fraction of realizable sequences undergoes two first-order transitions at the values 0 and 2 of the power-law exponent. We substantiate this finding by analytical reasoning and by a numerical method, proposed here, based on extreme value arguments, which can be applied to any given degree distribution. Our results reveal a fundamental reason why large scale-free networks without constraints on minimum and maximum degree must be sparse.Comment: 4 pages, 2 figure

Route to Topological Superconductivity via Magnetic Field Rotation

The verification of topological superconductivity has become a major experimental challenge. Apart from the very few spin-triplet superconductors with p-wave pairing symmetry, another candidate system is a conventional, two-dimensional (2D) s-wave superconductor in a magnetic field with a sufficiently strong Rashba spin-orbit coupling. Typically, the required magnetic field to convert the superconductor into a topologically non-trivial state is however by far larger than the upper critical field H_c2, which excludes its realization. In this article, we argue that this problem can be overcome by rotating the magnetic field into the superconducting plane. We explore the character of the superconducting state upon changing the strength and the orientation of the magnetic field and show that a topological state, established for a sufficiently strong out-of-plane magnetic field, indeed extends to an in-plane field orientation. We present a three-band model applicable to the superconducting interface between LaAlO_3 and SrTiO_3, which should fulfil the necessary conditions to realize a topological superconductor