SSH Model

One-Sentence Description

The SSH model is a one-dimensional tight-binding chain with alternating hopping amplitudes whose gapped phases are distinguished by a winding number when chiral symmetry is preserved.

Physical Setup

Each unit cell contains two sublattice orbitals, $A_j$ and $B_j$. The model keeps intracell hopping $v$ and intercell hopping $w$. It was introduced for polyacetylene and is now a standard first model of topological band theory.

Hamiltonian

A common real-space convention is

$$H = \sum_j \left( v\,a_j^\dagger b_j + w\,a_{j+1}^\dagger b_j + \mathrm{h.c.} \right).$$

With periodic boundary conditions, the Bloch Hamiltonian in the basis $(a_k,b_k)$ is

$$H(k) = d_x(k)\sigma_x+d_y(k)\sigma_y,$$

where

$$d_x(k)=v+w\cos(ka), \qquad d_y(k)=w\sin(ka).$$

The bands are

$$E_\pm(k) = \pm \sqrt{ v^2+w^2+2vw\cos(ka) }.$$

The gap closes when $\lvert v\rvert=\lvert w\rvert$ in this convention.

Symmetry and Topology

The ideal SSH Hamiltonian has chiral symmetry:

$$\sigma_z H(k)\sigma_z=-H(k).$$

When the band gap is open, the vector $(d_x(k),d_y(k))$ winds around the origin or does not wind around it as $k$ crosses the Brillouin zone. In the conventional termination, the phase with

$$\lvert w\rvert>\lvert v\rvert$$

has edge modes for an open chain, provided the boundary termination and symmetry assumptions match the topological convention.

What It Teaches

The SSH model teaches that topology can be visible in a simple one-dimensional band Hamiltonian. It also teaches that bulk topology, boundary termination, and protecting symmetry must be discussed together.

Canonical Links

• Symmetry-Protected Topological Phases places the free chiral-symmetric chain inside the broader many-body protection and stable-equivalence framework.
• Tight-Binding Model
• Tight-Binding Chain
• Berry Phase
• Zak Phase Preview
• Topological Invariants
• Chern Numbers
• Pauli-Matrix Hamiltonians

Common Mistakes

• Forgetting that the topological statement assumes an open gap and chiral symmetry.
• Confusing the intracell and intercell hopping convention when naming the topological phase.
• Assuming edge states appear for every boundary termination.
• Adding onsite sublattice imbalance and still using the same chiral winding argument without modification.
• Treating the SSH model as a two-dimensional Chern insulator; its basic invariant is one-dimensional.

Quick Check

Why does the SSH gap close when $\lvert v\rvert=\lvert w\rvert$?

Solution

The spectrum is $E_\pm(k)=\pm\sqrt{v^2+w^2+2vw\cos(ka)}$. For real $v$ and $w$, the square root can vanish only when the complex number $v+w e^{ika}$ vanishes. This requires equal magnitudes and the appropriate phase, so the gap closes at $\lvert v\rvert=\lvert w\rvert$.

References

• W. P. Su, J. R. Schrieffer, and A. J. Heeger, “Solitons in polyacetylene,” Physical Review Letters 42, 1698-1701, 1979.
• J. K. Asbóth, L. Oroszlány, and A. Pályi, A Short Course on Topological Insulators, Springer, 2016.
• B. A. Bernevig and T. L. Hughes, Topological Insulators and Topological Superconductors, Princeton University Press, 2013.