Counting fixed points of a finitely generated subgroup of Aff [C]

Given a finitely generated subgroup G of the group of affine transformations acting on the complex line C, we are interested in the quotient Fix(G)/G. The purpose of this note is to establish when this quotient is finite and in this case its cardinality. We give an application to the qualitative study of polynomial planar vector fields at a neighborhood of a nilpotent singular point.</p

Fraunhofer pattern in the presence of Majorana zero modes

Majorana bound states (MBSs) emerge as zero energy excitations in topological superconductors. At zero temperature, their presence gives a quantized conductance in NS junctions and a fractional Josephson effect in Josephson junctions when the parity is conserved. However, most of current experiments deviate from the theoretical predictions, yielding for example a non-quantized conductance or the absence of only few odd Shapiro steps. Although these results might be compatible with a topological ground state, it is also possible that a trivial scenario can mimic similar results, by means of accidental zero energy Andreev bound states (ZEABS) or simply by non-adiabatic transitions between trivial Andreev bound states. Here, we propose a new platform to investigate signatures of the presence of MBSs in the Fraunhofer pattern of Josephson junctions featuring quantum spin Hall edge states on the normal part and Majorana bound states at the NS interfaces. We use a tight-binding model to demonstrate a change in periodicity of the Fraunhofer pattern when comparing trivial and non-trivial regimes. We explain these results in terms of local and crossed Andreev bound states, which due to the spin-momentum locking, accumulate different magnetic flux and therefore become distinguishable in the Fraunhofer periodicity. Furthermore, we introduce a scattering model that captures the main results of the microscopic calculations with MBSs and extend our discussion to the main differences found using accidental ZEABS.Comment: 17 pages, 14 figures. Comments are welcom

Enhanced quasiparticle dynamics of quantum well states: the giant Rashba system BiTeI and topological insulators

In the giant Rashba semiconductor BiTeI electronic surface scattering with Lorentzian linewidth is observed that shows a strong dependence on surface termination and surface potential shifts. A comparison with the topological insulator Bi2Se3 evidences that surface confined quantum well states are the origin of these processes. We notice an enhanced quasiparticle dynamics of these states with scattering rates that are comparable to polaronic systems in the collision dominated regime. The Eg symmetry of the Lorentzian scattering contribution is different from the chiral (RL) symmetry of the corresponding signal in the topological insulator although both systems have spin-split surface states.Comment: 6 pages, 5 figure

On the Eigenvalue Density of Real and Complex Wishart Correlation Matrices

Wishart correlation matrices are the standard model for the statistical analysis of time series. The ensemble averaged eigenvalue density is of considerable practical and theoretical interest. For complex time series and correlation matrices, the eigenvalue density is known exactly. In the real case, however, a fundamental mathematical obstacle made it forbidingly complicated to obtain exact results. We use the supersymmetry method to fully circumvent this problem. We present an exact formula for the eigenvalue density in the real case in terms of twofold integrals and finite sums.Comment: 4 pages, 2 figure

Helical edge states in multiple topological mass domains

The two-dimensional topological insulating phase has been experimentally discovered in HgTe quantum wells (QWs). The low-energy physics of two-dimensional topological insulators (TIs) is described by the Bernevig-Hughes-Zhang (BHZ) model, where the realization of a topological or a normal insulating phase depends on the Dirac mass being negative or positive, respectively. We solve the BHZ model for a mass domain configuration, analyzing the effects on the edge modes of a finite Dirac mass in the normal insulating region (soft-wall boundary condition). We show that at a boundary between a TI and a normal insulator (NI), the Dirac point of the edge states appearing at the interface strongly depends on the ratio between the Dirac masses in the two regions. We also consider the case of multiple boundaries such as NI/TI/NI, TI/NI/TI and NI/TI/NI/TI.Comment: 11 pages, 15 figure

Helical edge states in multiple topological mass domains

The two-dimensional topological insulating phase has been experimentally discovered in HgTe quantum wells (QWs). The low-energy physics of two-dimensional topological insulators (TIs) is described by the Bernevig-Hughes-Zhang (BHZ) model, where the realization of a topological or a normal insulating phase depends on the Dirac mass being negative or positive, respectively. We solve the BHZ model for a mass domain configuration, analyzing the effects on the edge modes of a finite Dirac mass in the normal insulating region (soft-wall boundary condition). We show that at a boundary between a TI and a normal insulator (NI), the Dirac point of the edge states appearing at the interface strongly depends on the ratio between the Dirac masses in the two regions. We also consider the case of multiple boundaries such as NI/TI/NI, TI/NI/TI and NI/TI/NI/TI.Comment: 11 pages, 15 figure