Amoebas of algebraic varieties and tropical geometry

This survey consists of two parts. Part 1 is devoted to amoebas. These are images of algebraic subvarieties in the complex torus under the logarithmic moment map. The amoebas have essentially piecewise-linear shape if viewed at large. Furthermore, they degenerate to certain piecewise-linear objects called tropical varieties whose behavior is governed by algebraic geometry over the so-called tropical semifield. Geometric aspects of tropical algebraic geometry are the content of Part 2. We pay special attention to tropical curves. Both parts also include relevant applications of the theories. Part 1 of this survey is a revised and updated version of an earlier prepreint of 2001.Comment: 40 pages, 15 figures, a survey for the volume "Different faces in Geometry

Dirac fermion quantization on graphene edges: Isospin-orbit coupling, zero modes and spontaneous valley polarization

The paper addresses boundary electronic properties of graphene with a complex edge structure of the armchair/zigzag/armchair type. It is shown that the finite zigzag region supports edge bound states with discrete equidistant spectrum obtained from the Green's function of the continuum Dirac equation. The energy levels exhibit the coupling between the valley degree of freedom and the orbital quantum number, analogous to a spin-orbit interaction. The characteristic feature of the spectrum is the presence of a zero mode, the bound state of vanishing energy. It resides only in one of the graphene valleys, breaking spontaneously Kramers' symmetry of the edge states. This implies the spontaneous valley polarization characterized by the valley isospin $\pm 1/2$. The polarization is manifested by a zero-magnetic field anomaly in the local tunneling density of states, and is directly related to the local electric Hall conductivity.Comment: 9 pages, 6 figures, to be published in Phys. Rev.