02043nas a2200121   4500008004100000245006000041210006000101260001000161520154600171653008801717100002101805856009501826       2015                        en  d00aGeometric phases in graphene and topological insulators0 aGeometric phases in graphene and topological insulators  bSISSA3 aThis thesis collects three of the publications that the candidate produced during his Ph.D. studies. They all focus on geometric phases in solid state physics.

We first study topological phases of 2-dimensional periodic quantum systems, in absence of a spectral gap, like e.g. (multilayer) graphene. A topological invariant n_v in Z, baptized eigenspace vorticity, is attached to any intersection of the energy bands, and characterizes the local topology of the eigenprojectors around that intersection. With the help of explicit models, each associated to a value of n_v in Z, we are able to extract the decay at infinity of the single-band Wannier function w in mono- and bilayer graphene, obtaining |w(x)| <= const |x|^{-2} as |x| tends to infinity.

Next, we investigate gapped periodic quantum systems, in presence of time-reversal symmetry. When the time-reversal operator Theta is of bosonic type, i.e. it satisfies Theta^2 = 1, we provide an explicit algorithm to construct a frame of smooth, periodic and time-reversal symmetric (quasi-)Bloch functions, or equivalently a frame of almost-exponentially localized, real-valued (composite) Wannier functions, in dimension d <= 3. In the case instead of a fermionic time-reversal operator, satisfying Theta^2 = -1, we show that the existence of such a Bloch frame is in general topologically obstructed in dimension d=2 and d=3. This obstruction is encoded in Z_2-valued topological invariants, which agree with the ones proposed in the solid state literature by Fu, Kane and Mele.10aGeometric phases, graphene, topological insulators, Wannier functions, Bloch frames1 aMonaco, Domenico  uhttps://www.math.sissa.it/publication/geometric-phases-graphene-and-topological-insulators