Su-Schrieffer-Heeger (SSH) model
Let’s derive the edge states for a simple model: the Su-Schrieffer-Heeger (SSH) model. The SSH model is a one-dimensional lattice model that exhibits topological properties and edge states. It’s a great starting point to understand how edge states emerge in topological systems.
1. The SSH Model
The SSH model describes electrons hopping on a 1D chain with alternating hopping amplitudes \(t_1\) and \(t_2\). The Hamiltonian is:
\[H = \sum_{i} \left( t_1 c_{A,i}^\dagger c_{B,i} + t_2 c_{B,i}^\dagger c_{A,i+1} + \text{h.c.} \right),\]where:
- \(c_{A,i}^\dagger\) and \(c_{B,i}^\dagger\) create electrons on sublattices \(A\) and \(B\) at site \(i\),
- \(t_1\) and \(t_2\) are the intra-cell and inter-cell hopping amplitudes, respectively.
2. Bulk Hamiltonian in Momentum Space
To study the bulk properties, we Fourier transform the Hamiltonian into momentum space. Define the Fourier transforms:
\[c_{A,i} = \frac{1}{\sqrt{N}} \sum_k e^{ik \cdot i} c_{A,k}, \quad c_{B,i} = \frac{1}{\sqrt{N}} \sum_k e^{ik \cdot i} c_{B,k},\]where \(N\) is the number of unit cells, and \(k\) is the crystal momentum. The Hamiltonian becomes:
\[H = \sum_k \begin{pmatrix} c_{A,k}^\dagger & c_{B,k}^\dagger \end{pmatrix} H(k) \begin{pmatrix} c_{A,k} \\ c_{B,k} \end{pmatrix},\]where the bulk Hamiltonian \(H(k)\) is:
\[H(k) = \begin{pmatrix} 0 & t_1 + t_2 e^{-ik} \\ t_1 + t_2 e^{ik} & 0 \end{pmatrix}.\]This can be written as:
\[H(k) = \vec{d}(k) \cdot \vec{\sigma},\]where \(\vec{\sigma} = (\sigma_x, \sigma_y, \sigma_z)\) are the Pauli matrices, and:
\[\vec{d}(k) = (t_1 + t_2 \cos k, \, t_2 \sin k, \, 0).\]3. Topological Invariant
The SSH model is characterized by a topological invariant called the winding number \(W\), which is defined as:
\[W = \frac{1}{2\pi} \int_{-\pi}^{\pi} dk \, \frac{d\phi(k)}{dk},\]where \(\phi(k) = \arg(d_x(k) + i d_y(k))\). For the SSH model: If \(|t_1| < |t_2|\), \(W = 1\) (topologically non-trivial). If \(|t_1| > |t_2|\), \(W = 0\) (topologically trivial).
4. Edge States in the Topological Phase
When \(|t_1| < |t_2|\), the system is in the topological phase, and edge states appear at the boundaries of a finite chain. Let’s derive these edge states.
Finite Chain with Open Boundary Conditions
Consider a finite chain with \(N\) unit cells and open boundary conditions. The Hamiltonian is:
\[H = \sum_{i=1}^{N-1} \left( t_1 c_{A,i}^\dagger c_{B,i} + t_2 c_{B,i}^\dagger c_{A,i+1} + \text{h.c.} \right).\]We look for zero-energy eigenstates
\[|\psi \rangle = \sum_i (\psi_{A,i} c_{A,i}^\dagger + \psi_{B,i} c_{B,i}^\dagger) |0\rangle\]that satisfy
\[H|\psi \rangle = 0\]Recursion Relations
From the Schrödinger equation \(H|\psi \rangle = 0\), we get the recursion relations: \(t_1 \psi_{B,i} + t_2 \psi_{B,i-1} = 0, \quad t_1 \psi_{A,i} + t_2 \psi_{A,i+1} = 0.\)
Assume an exponential ansatz for the wavefunctions:
\[\psi_{A,i} = \alpha^i, \quad \psi_{B,i} = \beta^i.\]Substituting into the recursion relations, we get:
\[t_1 \beta^i + t_2 \beta^{i-1} = 0 \implies t_1 \beta + t_2 = 0 \implies \beta = -\frac{t_2}{t_1},\] \[t_1 \alpha^i + t_2 \alpha^{i+1} = 0 \implies t_1 + t_2 \alpha = 0 \implies \alpha = -\frac{t_1}{t_2}.\]Localization of Edge States
For \(|t_1| < |t_2|\), \(|\alpha| < 1\) and \(|\beta| > 1\). This means: \(\psi_{A,i}\) decays exponentially from the left edge (\(i = 1\)). \(\psi_{B,i}\) decays exponentially from the right edge (\(i = N\)).
Thus, the edge states are localized at the ends of the chain.
5. Summary
The SSH model exhibits edge states in the topological phase \(|t_1| < |t_2|\).
- These edge states are zero-energy modes localized at the ends of the chain.
- The existence of edge states is protected by the topological invariant (winding number).
This derivation illustrates how edge states emerge in a simple topological system. Similar principles apply to more complex models like 2D topological insulators and quantum Hall systems.