Generalization the model of Su-Schrieffer-Heeger (SSH)

Author: Davit Kereselidze
Keywords: Topological insulator, Phase diagram, Winding number, Dispersion curves, Hopping amplitudes
Annotation:

Theoretical and experimental researchs have shown that insulators divide into two groups, after 70 years from creation of quantum mechanics and solid state physics. One of these is “ordinary insulators” and others – “topological”. Valence band is fully occupied by electrons which belong to the first group of insulators, but conduction band is separated from valence band by energetic gap. Such kind of material is diamond. For the other group of insulators, energetic gap changes by the strong spin-orbital interaction. It means that electronic states of high energy are situated under the energetic gap and vice versa. For this reason, on the surface of the topological insulators, create many conduction states. They are the stable enough toward scattering on the ordinary defects. Even if exist high energetic barriers, surface states flow current. In this case we say that topological insulator is a crystalline material, which is conductor on the surface and it is insulator within the volume. It must be note that the number of charge carriers is constant in the topological insulators. It differs from ordinary insulators by this sign. The electronic band structure of non-interacting topological insulator within the volume looks like to the ordinary one. The Fermi’s level and the special surface states are dropped into the volume energetic gap and create the metallic surface. At 1987 firstly has started to study experimentally and theoretically topological insulators and it’s still timely. Such materials exist in the nature, for example Kawazulite mineral. Thereby, in the bulk part of the Bi-based 3D materials exist ordinary insulating or topological states (Nature Physics 2014). At 2012, several group of scientist suggest that samarium hexaboride is a topological insulator in accordance to the earlier theoretical predictions. Beside this, it’s possible to observe helical Dirac’s fermions in such structures and these quasiparticles are massless relativistic objects. There are highly popular above-mentioned crystalline structures in the modern solid state physics. For this reason, we’ve generalized such prominent example of topological insulators as Su-Schrieffer-Heeger’s (SSH) model. In this case, we’ve studied theoretically especially the long bulk part of quasi 1D chain by the four different hopping amplitudes. More concrete, we’ve derived analytically bulk matrix Hamiltonian in the momentum- space with its eigenfunction and dispersion relation. After this, we’ve constructed dispersion curves and evaluated conditions, which define the metallic phases from insulators. On the other hand, we speak about the phase transitions. We’ve defined the winding number for especially the long quasi 1D chain beside the above-mentioned things. Its every member of sum looks like to the same value expression of the SSH model outwardly. We can perfectly define any phase with this parameter. At last, we’ve constructed the phase diagrams with parametrization of hopping amplitudes and by calculation formula of winding number for the generalization model of SSH. As it defined, from one insulator phase transition to another is possible to cross metallic border in one case and via insulator border − on the other case.

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