Estudio del monopolo y dipolo eléctrico en el modelo tight-binding QTI (Quadrupole Topological Insulator) con cuadrupolo eléctrico fraccionalizado

Abstract

The macroscopic polarization is the most essential concept in any phenomenological description of dielectric materials but calculating this and other electric multipole moments in crystalline insulators is not an obvious process due to the quantum mechanical behavior of the electron cloud and the periodicity of the lattice.

In this project, we will study how the first three electric multipole moments are related to each other in crystalline insulators. For this purpose, we will use the tight-binding SSH (Su-Schrieffe-Heeger) model, that represents a one-dimensional crystalline insulator, and the tight-binding QTI (Quadrupole Topological Insulator) model, that represents a two-dimensional crystalline insulator.

We will verify that the one-dimensional model presents a quantized electric dipole moment that generates charge accumulations at the ends of the crystal, while the two-dimensional model presents a quantized electric quadrupole moment that generates charge accumulations at the corners of the crystal, as well as polarizations at its edges.

In order to compute the electric dipole moment, we will use the quantum mechanical operator of the position in one dimension, as well as an analogous operator in two dimensions to calculate the electric quadrupole moment.

Finally, to check the generality of the operators used in our algorithm, we will verify that the first three multipole moments in the QTI model behave as predicted by the theory, even when the quadrupole moment is not quantized.