Topology in the SSH Model

The Su–Schrieffer–Heeger (SSH) model is the minimal one-dimensional lattice model in which band topology can be formulated exactly. Its structure is a bipartite chain with two sites per unit cell and alternating hopping amplitudes $t_1$ and $t_2$. In real space, the Hamiltonian is

$$H=\sum_n \left(t_1 a_n^\dagger b_n+t_2 a_{n+1}^\dagger b_n+\mathrm{h.c.}\right).$$

After Fourier transformation, it becomes $H(k)=d_x(k)\sigma_x+d_y(k)\sigma_y$, with $d_x=t_1+t_2\cos k$ and $d_y=t_2\sin k$. The corresponding bands are $E_\pm(k)=\pm\sqrt{t_1^2+t_2^2+2t_1t_2\cos k}$. The absence of a $\sigma_z$ term gives the model chiral symmetry: with $\Gamma=\sigma_z$, one has ${\Gamma,H(k)}=0$. This constraint is the origin of the model’s topological classification.

Because chiral symmetry restricts the Bloch vector $\mathbf d(k)$ to the $d_x$-$d_y$ plane, the topology reduces to the geometry of a closed loop. Defining $q(k)=t_1+t_2e^{ik}$, the phase of $q(k)$ winds as $k$ traverses the Brillouin zone. The corresponding invariant is

$$\nu=\frac{1}{2\pi}\int_{-\pi}^{\pi} dk,\partial_k\arg q(k).$$

For $|t_1|>|t_2|$, the trajectory does not enclose the origin and $\nu=0$; for $|t_2|>|t_1|$, it winds once around the origin and $\nu=1$. Changing $\nu$ requires the loop to cross the origin, which occurs when $|t_1|=|t_2|$. At that point the bulk gap closes. The topological transition is therefore encoded directly in the singularity of the Bloch Hamiltonian.

The same distinction appears in the Berry phase of the occupied band. The Zak phase is $\gamma=i\int_{\mathrm{BZ}} dk,\langle u_-(k)|\partial_k u_-(k)\rangle$, and in the symmetry-protected SSH model it is quantized to $0$ or $\pi$ modulo $2\pi$. Through the modern theory of polarization, $P=-e\gamma/2\pi \pmod e$, so the two phases differ by $\Delta P=e/2 \pmod e$. This formulation gives the topology a direct real-space meaning: the transition moves the Wannier centre between the two inequivalent bond centres of the dimerized chain.

The bulk invariant becomes observable at a boundary. For an open chain in the regime $|t_2|>|t_1|$, a zero-energy edge state can live on a single sublattice. Its amplitudes satisfy $t_1\psi_n+t_2\psi_{n+1}=0$, giving $\psi_{n+1}=-(t_1/t_2)\psi_n$. The state is normalizable only when $|t_1/t_2|<1$, exactly the condition for the non-trivial winding phase. Its localization length obeys $\xi^{-1}=|\ln|t_2/t_1||$, which diverges as the bulk gap closes. Bulk topology and edge localization are therefore two manifestations of the same spectral structure.

The connection to polyacetylene becomes most transparent in the continuum limit. Near the gap-closing point, the SSH Hamiltonian reduces to a one-dimensional Dirac Hamiltonian,

$$H=-i\hbar v_F\sigma_y\partial_x+m(x)\sigma_x,$$

where the dimerization becomes the Dirac mass $m(x)$. The two possible dimerization patterns correspond to opposite signs of $m$. A domain wall satisfying $m(-\infty)m(+\infty)<0$ binds a Jackiw–Rebbi zero mode. In polyacetylene, this mode is the electronic origin of the soliton state and of the associated fractionalized charge. The fractionalization can equivalently be understood from the change in bulk polarization across the domain wall.

The SSH model also makes clear that topology is defined together with symmetry. Adding a staggered onsite potential introduces a term $\Delta\sigma_z$, breaks chiral symmetry, and allows the Bloch vector to leave the equatorial plane. The winding number then loses its protection, and the two dimerization patterns can be connected without closing the gap. Within the Altland–Zirnbauer classification, the ideal SSH chain belongs to class BDI and carries an integer invariant in one dimension.

Its enduring importance comes from this compression of topological band theory into a solvable model. Chiral symmetry constrains the Hamiltonian, the constraint permits a winding number, the winding controls polarization, and a change in bulk topology requires gap closing and produces boundary states. The SSH model therefore provides the simplest complete realization of symmetry-protected topology in a crystalline quantum system.