Problem on SSH Model Tight Binding Approach

[PhysicsTruth]

Homework Statement

Consider a one-dimensional chain of atoms as shown in the figure. Let the spacing between the atoms be $a$. Assume that the onsite energy is the same at each point and is equal to $0$ (without any loss of generality), but the hopping terms are of two types: $w$ denoted by a single bond and $v$ denoted by the double bond.

a) Write down the tight binding Hamiltonian.

b) Assume periodic boundary condition and find the band dispersion. Plot the band diagram for different choices of the parameters: $v > w$, $v = w$ and $v < w$. What do you observe?

c) Now find the eigenvectors which will give you the Bloch spinors $|u_{\pm}(k)>$. Calculate $A_{\pm}(k)=i<u_{\pm}(k)|\frac{d}{dk}u_{\pm}(k)>$ for the upper $(+)$ and lower $(-)$ bands. Then integrate over the Brillouin zone to find the winding number $g_{\pm} = -\frac{1}{\pi}\int A_{\pm}(k)dk$. Show that it is $0$ if $v > w$ but equal to $1$ if $v < w$ and they do not
depend on the exact values of the parameters, i.e., it is an invariant (topological).

d) The bulk-boundary correspondence says that if the invariant is nonzero, in a finite system, there will be a zero mode that will be protected. Let’s check this. Redo the tight-binding calculation but now in real space. Assume there are 20 lattice points and find the energies – this has to be done numerically (diagonalize
a $20 \times 20$ matrix). Try different values of the parameters. Show that as long as $v < w$, it does not matter,
there is always an energy level at zero value.

e) On which lattice point does the zero mode have maximum weight? Compare it with some other energy level that is away from zero.

Relevant Equations

No such relevant equations.

I'd like to proceed in a linear fashion, taking each part on one by one. For the first part, we can write the Hamiltonian as $H = \sum_{n}^{N} w(c_{An}^{\dagger}c_{Bn}+c_{Bn}^{\dagger}c_{An})+v(c_{Bn}^{\dagger}c_{A(n+1)}+c_{A(n+1)}^{\dagger}c_{Bn})$. We can convert the creation and annihilation operators to their momentum space analogues to get - $a_{k} = \frac{1}{\sqrt{N}} \sum _{n}^{N}e^{-ikna}c_{An}$ and $b_{k} = \frac{1}{\sqrt{N}} \sum _{n}^{N}e^{-ikna}c_{Bn}$. Using this, we get, $H = \sum _{k} w(a_{k}^{\dagger}b_{k}+b_{k}^{\dagger}a_{k})+v(e^{ika}b_{k}^{\dagger}a_{k}+e^{-ika}a_{k}^{\dagger}b_{k})$. Taking the spinors $\psi_k = \begin{pmatrix} a_{k} \\ b_{k} \end{pmatrix}$, we have - $H = \sum _{k} \psi_k^{\dagger} \begin{pmatrix} 0 & w+ve^{-ika} \\ w+ve^{ika} & 0 \end{pmatrix} \psi_k$.

For the second part, we diagonalise the matrix in order to find the dispersion relation as $E = \sqrt{v^2+w^2+2vwcos(ka)}$.

It'd be helpful if someone could confirm that these calculations are indeed correct, so that I can move on to the later parts.

 

[member38644]

Yes, your calculations for the first and second parts are correct. The Hamiltonian and dispersion relation you have derived are commonly used in the SSH model and are an important starting point for understanding the properties of the system. Good job!