It's not so difficult to imagine it. The topological insulator is behaving with respect to movable electrons like the hydrophobic sponge with respect to mercury: the electrons are expelled from its pores and they collect at its surface, when they form a conductive layer. Because of it the electrons are mutually compressed each other and their repulsive forces are compensated in certain points, so that the electrons transfer their charge freely there in similar way, like the electrons within superconductors or at the surface of graphene. The renormalization is mathematical procedure used in quantum field theories for reconciliation of mutually inconsistent extrinsic perspective of quantum mechanics and intrinsic perspective of general relativity. At the case of topological insulators the renormalization means that the electrons are moving so chaotically that their collective behavior from intrinsic perspective becomes blurred with their behavior from extrinsic perspective.
The phase transition of electrons compressed at the surface of the topological insulator can be simulated easily both physically, both at the computer with array of particles which are mutually repulsing at the large distance (so-called plasma crystal). When the mutual compression of particles will exceed certain limit, then the particles will form a chaotic fluid, which will exhibit a foamy density fluctuations (in similar way, like the virtual particles inside of vacuum in accordance to dense aether model). So instead of clusters of electrons with positive curvature we suddenly have the foamy density fluctuations of the opposite curvature - a topological inversion of system happens there and the low-dimensional formal solutions of system from both sides will become singular.
One the option how to solve such a situation numerically is to use the hyperdimensional model, which suffers with high number of hidden parameters and low stability. The renormalization procedure just helps to cross this "critical point" without resorting to poorly conditioned hyperdimensional models. It's based on the fact, that whereas the function describing the behavior of system from both sides of singularity diverge, their higher derivations limit to singular point more smoothly - so they can be used for finding of common averaged solution. The wild behavior of physical model at the singular point will get blurred and smoothed in this way. The renormalized solution is sorta unphysical in this way, but it helps to apply the low-dimensional models to the high-dimensional systems smoothly.
Note that at the above simulation the particles tend to form a hexagonal mesh. The distribution of electrons at the surface of topological insulator therefore follows a hexagonal Mott lattice around singular points, which is easier to imagine in 3D. The existence of hexagonal distribution of electrons at the surface can be therefore analyzed with polarized light shinning at it under low angle (ARPES method), which will release the photoelectrons just at the moment, when the plane of polarization of light will become collinear with the orientation of Mott lattice.
"The wild behavior of physical model at the singular point will get blurred and smoothed in this way. The renormalized solution is sorta unphysical in this way". As is the Infinite Improbability Drive from The Hitchhiker's Guide to the Galaxy.
Best example of what renormalization does and cannot do is the reconciliation of relativity and quantum mechanics. These theories are describing abstract four-dimensional multiverses, which are separated with many dimensions each other. You should be able to derive the trees, bees, grass and human beings with renormalization, which of course isn't possible, until you working in four-dimensional space-time.
Argiod
Feb 8, 2013