Problem on SSH Model Tight Binding Approach

In summary, the conversation discusses the process of converting creation and annihilation operators to their momentum space analogues and using them to write the Hamiltonian as a sum. The spinors are then used to diagonalize the matrix and find the dispersion relation. Confirmation of the correctness of the calculations is requested before proceeding to later parts.
  • #1
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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.
 
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  • #2


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!
 

Related to Problem on SSH Model Tight Binding Approach

What is the SSH model in the context of the tight-binding approach?

The Su-Schrieffer-Heeger (SSH) model is a one-dimensional tight-binding model used to describe the behavior of electrons in a polymer chain, such as polyacetylene. It considers a chain with alternating strong and weak bonds between neighboring atoms, leading to a dimerized structure. The model is essential for studying topological properties and solitons in condensed matter physics.

How does the SSH model demonstrate topological phases?

The SSH model exhibits topological phases through its band structure, which can be characterized by a topological invariant known as the winding number. Depending on the relative strengths of the alternating bonds, the model can be in a topologically trivial phase (winding number zero) or a non-trivial phase (winding number one). The non-trivial phase is associated with edge states that appear at the boundaries of the chain, which are robust against certain types of perturbations.

What are the key parameters in the SSH model?

The key parameters in the SSH model are the hopping amplitudes between neighboring sites. Typically, these are denoted as \( t \) for the intra-cell (strong) hopping and \( t' \) for the inter-cell (weak) hopping. The ratio \( t/t' \) determines the topological phase of the system. Additionally, the model can include on-site energy terms and disorder to study various physical phenomena.

How do you calculate the band structure in the SSH model?

To calculate the band structure in the SSH model, you start by writing down the tight-binding Hamiltonian, which includes the hopping terms between neighboring sites. By applying a Fourier transform, you convert the Hamiltonian into momentum space, resulting in a 2x2 matrix for each wavevector \( k \). Diagonalizing this matrix gives the energy eigenvalues, which form the band structure. The dispersion relation typically shows two bands that can touch at the Brillouin zone boundaries, depending on the hopping parameters.

What are the implications of edge states in the SSH model?

Edge states in the SSH model are a direct consequence of its topological nature. In the topologically non-trivial phase, zero-energy edge states appear at the ends of the chain when it is open. These states are localized at the boundaries and are protected by the topological properties of the bulk. This means they are robust against perturbations that do not close the bulk band gap, making them of interest for applications in quantum computing and robust electronic devices.

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