Fast Car Physics,by Chuck Edmondson

An introduction to the theory of modular symmetries in two-dimensional materials, and its application to ‘relativistic’ group IV materials like graphene, silicene, germanene and stanene, is given. Universal properties of the magneto-electric Hall effect are extracted by projecting experimental transport data directly onto the phase diagram. When families of data depending on the dominant scale parameter (usually temperature) are available, we can extract flow lines that chart the geometry of the phase diagram, including the location of quantum critical points and phase boundaries connecting these. The universal data are used to identify emergent modular symmetries, which are infinite discrete groups of fractional linear (Möbius) transformations. Such symmetries are extremely rigid, and therefore spawn a host of sharp predictions that are easy to falsify, but so far they have failed to fail. The unique topology of the Fermi surface in the graphene family gives a robust gapless mode with linear dispersion (relativistic Dirac cones) that shifts the spectrum of Landau levels that appear when the material is placed in a strong magnetic field. The modular analysis can be extended to this case, and where reliable data are available, there appears to be agreement. A convincing case for the ‘relativistic’ quantum Hall group is hampered by the paucity of fractional quantum Hall data, the absence of scaling data and the crossover between different scaling regimes. This is likely to change in the near future, as scaling data for graphene are just now becoming available.