# How to calculate Edge states of Topological insulators

Topological insulators are novel state of matter in which bulk is insulator and edges are gapless. How do we calculate these gapless states? I am reading the following PRL

Feng Liu and Katsunori Wakabayashi, Novel Topological Phase with a Zero Berry Curvature The pdf file is available here.

In this PRL they have plotted gapless states in figure.3 but they are not mentioning about how they plotted these states. Is there any simple model from which I can understand how to plot ribbon spectra of edges states (as they have done in this paper).
They are providing a Hamiltonian in last paragraph on page#1 from which I successfully plotted the energy band structure given in Figure.1 using MATLAB.

• One way of plotting the edge spectra is to consider Eq. 1 and perform a Fourier transformation along one direction, say(x), assuming periodic boundary condition, and treat another direction with open boundary. Then calculate the spectrum of this Hamiltonian with $k_{x}$. A nice example would be 2D quantum Hall effect in lattice rhodia.ph.tsukuba.ac.jp/~hatsugai/modules/pico/PDF./Riemann.pdf – user123 Nov 6 '17 at 1:57

It is quite straigthforward. First you write your hamiltonian in second quantization formalism in crsytal momentum basis. Then you make a fourier transform to fermionic field operators such that, now they are in lattice site basis. Now your hamiltonian is in form of $$\mathcal{H}=\sum_{i,j,\mathbf{k_{\perp}}}\Psi^{\dagger}_{j,\mathbf{k_{\perp}}}H_{i,j,\mathbf{k_{\perp}}}\Psi_{i,\mathbf{k_{\perp}}}$$ And if you find the eigen values of $H_{i,j,\mathbf{k_{\perp}}}$ with $ij$ determined by the size of your finite system the eigen values will look like the figure in your example.