Invariância de escala no marcador local de Chern para isolantes topológicos de Anderson

Abstract

In this Master Dissertation, we investigate the phase transitions induced by Anderson’s impurities in topological insulators. Topological insulators are crystals with an insulating band structure, that is, with a gap at the Fermi energy, but with helical conducting states at the edges. The topological insulators are characterized via their band ordering and topological invariants. Particularly, we are interested in the Chern number. One calls the band ordering as topologically non-trivial whenever the Chern number is a finite integer, and trivial if it is zero. The transition between these regimes occurs via band inversions near the Fermi level. This inversion may be induced by spin-orbit coupling, electric fields, etc. Here we are interested in the transitions induced by Anderson’s impurities, a mechanism that leads to topological Anderson insulators (TAI). These impurities break the translational symmetry of the crystal and do not allow for a characterization in terms of the Chern number, which is defined in the quasimomentum space exploring the translational invariance. Therefore, here we review and the local Chern marker, which corresponds to a representation of the Chern number in coordinates space. Interestingly, while the Chern number is always an integer, the local marker goes smoothly from zero to integer values throughout the phase transition. This feature allow us to search for universal functions and scaling invariance of the local marker as a function of the impurity strength W and system size L × L. Our results show that the local Chern marker scales as C0 ≡ C0([W − Wc] · L1/μ), where Wc is the fix point and μ is the critical exponent. For topological insulators defined by the BHZ model, we find μ ≈ 2. Additionally, we investigate other characteristics of TAIs for its different phases: conductance, local density of states, and wave-functions. We have calculated its phase diagram as a function of the the BHZ mass M and impurity strength W to identify four phases: trivial, TI, TAI, and trivial Anderson localization. We use the kwant python package to build the Hamiltonian and calculate the conductance, wave functions, and local density of states. For the local Chern number, we have implemented an approximation via the Kernel Polynomial Method. All quantities that depend upon disorder are obtained as averages over an ensemble of random samples.