Topological Majorana and Dirac zero modes in superconducting vortex cores

We provide an argument based on flux insertion to show that certain superconductors with a non-trivial topological invariant have protected zero modes in their vortex cores. This argument has the flavor of a two dimensional index theorem and applies to disordered systems as well. It also provides a new way of understanding the zero modes in the vortex cores of a spinless $p_{x} + i p_{y}$ superconductor. Applying this approach to superconductors with and without time reversal and spin rotational symmetry, we predict the necessary and sufficient conditions for protected zero modes to exist in their vortices.Comment: 5 pages, 1 figur

A six-factor asset pricing model

The present study introduce the human capital component to the Fama and French five-factor model proposing an equilibrium six-factor asset pricing model. The study employs an aggregate of four sets of portfolios mimicking size and industry with varying dimensions. The first set consists of three set of six portfolios each sorted on size to B/M, size to investment, and size to momentum. The second set comprises of five index portfolios, third, a four-set of twenty-five portfolios each sorted on size to B/M, size to investment, size to profitability, and size to momentum, and the final set constitute thirty industry portfolios. To estimate the parameters of six-factor asset pricing model for the four sets of variant portfolios, we use OLS and Generalized method of moments based robust instrumental variables technique (IVGMM). The results obtained from the relevance, endogeneity, overidentifying restrictions, and the Hausman's specification, tests indicate that the parameter estimates of the six-factor model using IVGMM are robust and performs better than the OLS approach. The human capital component shares equally the predictive power alongside the factors in the framework in explaining the variations in return on portfolios. Furthermore, we assess the t-ratio of the human capital component of each IVGMM estimates of the six-factor asset pricing model for the four sets of variant portfolios. The t-ratio of the human capital of the eighty-three IVGMM estimates are more than 3.00 with reference to the standard proposed by Harvey et al. (2016). This indicates the empirical success of the six-factor asset-pricing model in explaining the variation in asset returns

On the Z2Z_2 classification of Quantum Spin Hall Models

We propose an alternative formulation of the $Z_2$ topological index for quantum spin Hall systems and band insulators when time reversal invariance is not broken. The index is expressed in terms of the Chern numbers of the bands of the model, and a connection with the number of pairs of robust edge states is thus established. The alternative index is easy to compute in most cases of interest. We also discuss connections with the recently proposed spin Chern number for quantum spin Hall models.Comment: Presentation changed to improve clarity, some technical aspects of the topological arguments including material previously cited as unpublished notes have now been added as an appendi

Abelian Floquet symmetry-protected topological phases in one dimension

Time-dependent systems have recently been shown to support novel types of topological order that cannot be realised in static systems. In this paper, we consider a range of time-dependent, interacting systems in one dimension that are protected by an Abelian symmetry group. We classify the distinct topological phases that can exist in this setting and find that they may be described by a bulk invariant associated with the unitary evolution of the closed system. In the open system, nontrivial phases correspond to the appearance of edge modes in the many-body quasienergy spectrum, which relate to the bulk invariant through a form of bulk-edge correspondence. We introduce simple models which realise nontrivial dynamical phases in a number of cases, and outline a loop construction that can be used to generate such phases more generally.Comment: 13 pages, 1 figure; Published versio