Depth, Highness and DNR degrees

We study Bennett deep sequences in the context of recursion theory; in particular we investigate the notions of O(1)-deepK, O(1)-deepC , order-deep K and order-deep C sequences. Our main results are that Martin-Loef random sets are not order-deepC , that every many-one degree contains a set which is not O(1)-deepC , that O(1)-deepC sets and order-deepK sets have high or DNR Turing degree and that no K-trival set is O(1)-deepK.Comment: journal version, dmtc

Topological aspects of poset spaces

We study two classes of spaces whose points are filters on partially ordered sets. Points in MF spaces are maximal filters, while points in UF spaces are unbounded filters. We give a thorough account of the topological properties of these spaces. We obtain a complete characterization of the class of countably based MF spaces: they are precisely the second-countable T_1 spaces with the strong Choquet property. We apply this characterization to domain theory to characterize the class of second-countable spaces with a domain representation.Comment: 29 pages. To be published in the Michigan Mathematical Journa

Splitting of the Zero-Energy Landau Level and Universal Dissipative Conductivity at Critical Points in Disordered Graphene

We report on robust features of the longitudinal conductivity ($\sigma_{xx}$) of the graphene zero-energy Landau level in presence of disorder and varying magnetic fields. By mixing an Anderson disorder potential with a low density of sublattice impurities, the transition from metallic to insulating states is theoretically explored as a function of Landau-level splitting, using highly efficient real-space methods to compute the Kubo conductivities (both $\sigma_{xx}$ and Hall $\sigma_{xy}$). As long as valley-degeneracy is maintained, the obtained critical conductivity $\sigma_{xx}\simeq 1.4 e^{2}/h$ is robust upon disorder increase (by almost one order of magnitude) and magnetic fields ranging from about 2 to 200 Tesla. When the sublattice symmetry is broken, $\sigma_{xx}$ eventually vanishes at the Dirac point owing to localization effects, whereas the critical conductivities of pseudospin-split states (dictating the width of a $\sigma_{xy}=0$ plateau) change to $\sigma_{xx}\simeq e^{2}/h$, regardless of the splitting strength, superimposed disorder, or magnetic strength. These findings point towards the non dissipative nature of the quantum Hall effect in disordered graphene in presence of Landau level splitting

On Gravity, Torsion and the Spectral Action Principle

We consider compact Riemannian spin manifolds without boundary equipped with orthogonal connections. We investigate the induced Dirac operators and the associated commutative spectral triples. In case of dimension four and totally anti-symmetric torsion we compute the Chamseddine-Connes spectral action, deduce the equations of motions and discuss critical points.Comment: minor modifications, some further typos fixe

On Martin's Pointed Tree Theorem

We investigate the reverse mathematics strength of Martin's pointed tree theorem (MPT) and one of its variants, weak Martin's pointed tree theorem (wMPT)

Efficient Linear Scaling Approach for Computing the Kubo Hall Conductivity

We report an order-N approach to compute the Kubo Hall conductivity for disorderd two-dimensional systems reaching tens of millions of orbitals, and realistic values of the applied external magnetic fields (as low as a few Tesla). A time-evolution scheme is employed to evaluate the Hall conductivity $\sigma_{xy}$ using a wavepacket propagation method and a continued fraction expansion for the computation of diagonal and off-diagonal matrix elements of the Green functions. The validity of the method is demonstrated by comparison of results with brute-force diagonalization of the Kubo formula, using (disordered) graphene as system of study. This approach to mesoscopic system sizes is opening an unprecedented perspective for so-called reverse engineering in which the available experimental transport data are used to get a deeper understanding of the microscopic structure of the samples. Besides, this will not only allow addressing subtle issues in terms of resistance standardization of large scale materials (such as wafer scale polycrystalline graphene), but will also enable the discovery of new quantum transport phenomena in complex two-dimensional materials, out of reach with classical methods.Comment: submitted PRB pape