Su-Schrieffer-Heeger (SSH) model | iℏ∮dͩ𝛑•💕

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} (t_{1} c_{A, i}^{†} c_{B, i} + t_{2} c_{B, i}^{†} c_{A, i + 1} + h.c.),

where:

$c_{A, i}^{†}$ and $c_{B, i}^{†}$ 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^{i k \cdot i} c_{A, k}, c_{B, i} = \frac{1}{\sqrt{N}} \sum_{k} e^{i k \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{matrix} c_{A, k}^{†} & c_{B, k}^{†} \end{matrix}) H (k) (\begin{matrix} c_{A, k} \\ c_{B, k} \end{matrix}),

where the bulk Hamiltonian $H (k)$ is:

H (k) = (\begin{matrix} 0 & t_{1} + t_{2} e^{- i k} \\ t_{1} + t_{2} e^{i k} & 0 \end{matrix}) .

This can be written as:

H (k) = \vec{d} (k) \cdot \vec{σ},

where $\vec{σ} = (σ_{x}, σ_{y}, σ_{z})$ 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 π} \int_{- π}^{π} d k \frac{d ϕ (k)}{d k},

where $ϕ (k) = \arg (d_{x} (k) + i d_{y} (k))$ . 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} (t_{1} c_{A, i}^{†} c_{B, i} + t_{2} c_{B, i}^{†} c_{A, i + 1} + h.c.) .

We look for zero-energy eigenstates

| ψ ⟩ = \sum_{i} (ψ_{A, i} c_{A, i}^{†} + ψ_{B, i} c_{B, i}^{†}) | 0 ⟩

that satisfy

H | ψ ⟩ = 0

Recursion Relations

From the Schrödinger equation $H | ψ ⟩ = 0$ , we get the recursion relations: $t_{1} ψ_{B, i} + t_{2} ψ_{B, i - 1} = 0, t_{1} ψ_{A, i} + t_{2} ψ_{A, i + 1} = 0.$

Assume an exponential ansatz for the wavefunctions:

ψ_{A, i} = α^{i}, ψ_{B, i} = β^{i} .

Substituting into the recursion relations, we get:

t_{1} β^{i} + t_{2} β^{i - 1} = 0 ⟹ t_{1} β + t_{2} = 0 ⟹ β = - \frac{t_{2}}{t_{1}},

t_{1} α^{i} + t_{2} α^{i + 1} = 0 ⟹ t_{1} + t_{2} α = 0 ⟹ α = - \frac{t_{1}}{t_{2}} .

Localization of Edge States

For $| t_{1} | < | t_{2} |$ , $| α | < 1$ and $| β | > 1$ . This means: $ψ_{A, i}$ decays exponentially from the left edge ( $i = 1$ ). $ψ_{B, i}$ 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.