Faddeev LeVerrier Algorithmm for a Chern Insulator

Miguel Alvarado and Alfredo Levy Yeyati

Departamento de Fisica Teorica de la Materia Condensada C-V, Condensed Matter Physics Center (IFIMAC) and Instituto Nicolas Cabrera, Universidad Autonoma de Madrid, E-28049 Madrid, Spain

1. Project Description

In this work we introduce a general method to obtain the boundary Green's function (bGF) of topological 2D Hamiltonians taking advantage of the numerical Faddeev-LeVerrier algorithm (FLA).

The main idea behind the bGF method is to compute the GF on the open boundary of a (d-1) semi-infinite system starting from a d-dimensional infinite bulk. To break the translational symmetry of the infinite system in the direction perpendicular to the boundary a (d-1) surface infinite impurity potential is induced. The momenta in the direction parallel to the boundary is preserved. 

The main input to compute the bGF using the FLA is the polynomial decomposition of the Hamiltonian for a given set of parallel momenta (k) to the junction and frequencies (w).

Using FLA we obtain the auxiliary matrix to compute the adjugate of the secular equation M(k) and the coefficients of the characteristic polynomial C(k). From C(K) we can compute the characteristic polynomial P(k,w) for any desired frequency and solve it to obtain the roots z(k,w).

Both z(k,w) and M(k) are the key ingredients to compute the unperturbed GFs in real space using the residue theorem. Finally, we use Dyson equation to compute the bGFs of the system from the unperturbed ones.

We illustrate our formalism analyzing the edge features like the spectral density of the bulk and boundaries of a 2D Chern insulator.

See arXiv:2107.10195 for a detailed description of the method.

2. Table of Contents

i) main_GF.m : main code to plot the spectral density of the open boundary of the Chern Insulator. 

ii) fun_FFA.m : rutine to compute the FLA from the polynomial decomposition of the Chern Insulator Hamiltonian to obtain the adjugate of the secular equation M(k) and the coefficients of the characteristic polynomial C(k)

iii) fun_polynomial.m : rutine to compute the characteristic polynomial P(k,w) from its the coefficients C(k) 

iv) fun_residues.m : rutine to compute local GFs using the residue theorem from the roots of the characteristic polynomial z(k,w) and the adjugate of the secular equation M(k)

3. How to use 

Run the main_GF.m file to obtain the spectral density of one of the boundaries of the Chern Insulator. If the term Mass < 2 the system is in the topological nontrivial regime hosting chiral edge states.

The codes are compatible with Octave