nLab Su-Schrieffer-Heeger model

Idea

The Su-Schrieffer-Heeger model (SSH) is a simple spin chain-like model for a type of 1-dimensional topological insulator quantum material. In theoretical physics it mainly serves as an instructive toy model example of topological phases of matter, while experimentally it gives an approximate description of the electron band-structure of materials like molecular trans-polyacetylene chains (which was the original motivation of Su, Schrieffer & Heeger 1979.

As a lattice model, SSH has a unit cell with two sites, traditionally denoted “$A$” and “$B$”, hence with a 2-dimensional (2 “band”) Hilbert space $\mathscr{H}_n \simeq \mathbb{C}^2 \simeq \mathbb{C}\langle A, B\rangle$ at each site, and the Hamiltonian operator is the sum of

1. an intracell hopping term between sites $A$ and $B$ in the same cell

2. an extracell hopping term between the $B$-site of one cell with the $A$-site of the next one.

The parameter space of the model is that of the coefficients of these two terms:

• $v \in \mathbb{R}_{\gt 0}$, the intracell hopping strength,

• $w \in \mathbb{R}_{\gt 0}$, the intercell hopping strength.

After Fourier transform, the Bloch Hamiltonian of the SSH model

$$H = \textstyle{\int_{[k] \in S^1}} H_{k} \;\in\; End\Big( \textstyle{\int_{k \in S^1}} \mathbb{C}^2 \Big)$$

is given (up to arbitrary energy shift and unitary transformation) by (cf. Asbóth, Oroszlány & Pályi 2016 (1.18), Batra & Sheet 2020):

(1)$$\begin{aligned} [k] &\mapsto H_{[k]} \\ &\coloneqq \big( v + w \cdot cos(k) \big) \sigma_x + \big( w \cdot sin(k) \big) \sigma_y \,, \end{aligned}$$

where $\sigma_{(-)} \in End(\mathbb{C}^2)$ denotes the Pauli matrices.

To note here that these Bloch Hamiltonians $H_{[k]}$ (1)

1. have no contribution of the Pauli matrix $\sigma_z$, which more abstractly means that they satisfy “chiral symmetry” in the sense that

   (2)$$\sigma_z H_{[k]} \sigma_z = - H_{[k]} \,,$$

2. generically have two real eigenvalues of opposite sign (labeling the valence band of negative energy and the conduction band of positive energy, respectively), except when $H_{[k]} = 0$, where the energy gap between these eigenvalues closes, which happens when $v = w$.

Hence for $v = w$ the SSH model describes an ordinary conductor and for $v \neq w$ an insulator.

In the latter case there are still two qualitatively different regimes: When $v \lt w$, then the Bloch Hamiltonian (1) regarded as a map to the subspace of gapped and chiral (2) 2-band Hamiltonians

$$\begin{aligned} H_{(-)} \colon S^1 & \longrightarrow End(\mathbb{C}^2)_{gap, chr} \\ & \coloneqq \left\{ H \in End(\mathbb{C}^2) \,\bigg\vert\, \substack{ H^\dagger = H,\, \sigma_z H \sigma_z = - H \\ Eig_{\gt 0}(H) \simeq \mathbb{C} ,\, Eig_{\lt 0}(H) \simeq \mathbb{C} } \right\} \\ & \simeq \mathbb{R}^2 \setminus \{0\} \\ & \underset{hmpty}{\simeq} S^1 \end{aligned}$$

has a non-trivial homotopy class and hence describes a topological insulator phase, namely it has Hopf degree (“winding number”) equal to $\pm 1$ (depending on conventions); while for $v \gt w$ the Hopf degree winding number vanishes, which hence describes an ordinary insulator phase.

Related entries

• Haldane model

References

The original article:

• W. P. Su, J. R. Schrieffer, A. J. Heeger: Solitons in Polyacetylene, Phys. Rev. Lett. 42 (1979) 1698 [doi:10.1103/PhysRevLett.42.1698]

Review:

• János K. Asbóth, László Oroszlány, András Pályi, Chapter 1 of: A Short Course on Topological Insulators: Band-structure topology and edge states in one and two dimensions, Lecture Notes in Physics 919, Springer (2016) [arXiv:1509.02295, doi:10.1007/978-3-319-25607-8]

• Navketan Batra, Goutam Sheet: Understanding Basic Concepts of Topological Insulators Through Su-Schrieffer-Heeger (SSH) Model, Resonance 25 (2020) 765-786 [arXiv:1906.08435, doi:10.1007/s12045-020-0995-x]

• Weibo Xu: The Su-Schrieffer-Heeger model on a one-dimensional lattice: Analytical wave functions of topological edge states [arXiv:2509.16708]

See also:

• Wikipedia: Su–Schrieffer–Heeger model

Realization in topological photonics:

• Shiqi Xia, Ziteng Wang, Domenico Bongiovanni, Dario Jukić, Daohong Song, Liqin Tang, Jingjun Xu, Roberto Morandotti, Hrvoje Buljan, Zhigang Chen: Topological photonics in one-dimensional settings [arXiv:2602.08011]