Dirac Operator Zero-modes on a Torus

We study Dirac operator zero-modes on a torus for gauge background with uniform field strengths. Under the basic translations of the torus coordinates the wave functions are subject to twisted periodic conditions. In a suitable torus coordinates the zero-mode wave functions can be related to holomorphic functions of the complex torus coordinates. We construct the zero-mode wave functions that satisfy the twisted periodic conditions. The chirality and the degeneracy of the zero-modes are uniquely determined by the gauge background and are consistent with the index theorem.Comment: 28 pages, 2 figure

Graphene: new bridge between condensed matter physics and quantum electrodynamics

Graphene is the first example of truly two-dimensional crystals - it's just one layer of carbon atoms. It turns out to be a gapless semiconductor with unique electronic properties resulting from the fact that charge carriers in graphene demonstrate charge-conjugation symmetry between electrons and holes and possess an internal degree of freedom similar to ``chirality'' for ultrarelativistic elementary particles. It provides unexpected bridge between condensed matter physics and quantum electrodynamics (QED). In particular, the relativistic Zitterbewegung leads to the minimum conductivity of order of conductance quantum $e^2/h$ in the limit of zero doping; the concept of Klein paradox (tunneling of relativistic particles) provides an essential insight into electron propagation through potential barriers; vacuum polarization around charge impurities is essential for understanding of high electron mobility in graphene; index theorem explains anomalous quantum Hall effect.Comment: misprints are fixed; to appear in special issue of Solid State Communication

Fermion Wavefunctions in Magnetized branes: Theta identities and Yukawa couplings

Computation of Yukawa couplings, determining superpotentials as well as the K\"{a}hler metric, with oblique (non-commuting) fluxes in magnetized brane constructions is an interesting unresolved issue, in view of the importance of such fluxes for obtaining phenomenologically viable models. In order to perform this task, fermion (scalar) wavefunctions on toroidally compactified spaces are presented for general fluxes, parameterized by Hermitian matrices with eigenvalues of arbitrary signatures. We also give explicit mappings among fermion wavefunctions, of different internal chiralities on the tori, which interchange the role of the flux components with the complex structure of the torus. By evaluating the overlap integral of the wavefunctions, we give the expressions for Yukawa couplings among chiral multiplets arising from an arbitrary set of branes (or their orientifold images). The method is based on constructing certain mathematical identities for general Riemann theta functions with matrix valued modular parameter. We briefly discuss an application of the result, for the mass generation of non-chiral fermions, in the SU(5) GUT model presented by us in arXiv:0709.2799.Comment: 77 pages, v2:Some additions and improvements in text, version to appear in Nucl. Phys.

The Chiral Magnetic Effect and Axial Anomalies

We give an elementary derivation of the chiral magnetic effect based on a strong magnetic field lowest-Landau-level projection in conjunction with the well-known axial anomalies in two- and four-dimensional space-time. The argument is general, based on a Schur decomposition of the Dirac operator. In the dimensionally reduced theory, the chiral magnetic effect is directly related to the relativistic form of the Peierls instability, leading to a spiral form of the condensate, the chiral magnetic spiral. We then discuss the competition between spin projection, due to a strong magnetic field, and chirality projection, due to an instanton, for light fermions in QCD and QED. The resulting asymmetric distortion of the zero modes and near-zero modes is another aspect of the chiral magnetic effect.Comment: 33 pages, 5 figures, to appear in Lect. Notes Phys. "Strongly interacting matter in magnetic fields" (Springer), edited by D. Kharzeev, K. Landsteiner, A. Schmitt, H.-U. Ye