Berry phase in 1D materials

The Berry phase $\phi_B$ is the phase that an eigenstate acquires after its momentum vector goes around a circle at constant energy around the Dirac point.

It is defined as $\phi_B = -i \int \langle\psi|\partial_{\theta}|\psi\rangle$ and is well-known to be non-trivial in 2D material graphene, where the eigenstate is $\psi = \left(1, e^{i \theta} \right)^T$ and so $\phi_B = \pi$.

What is the physical meaning of Berry phase in 1D material? How to go around a circle in 1D?

The Berry phase in one dimension is usually called the Zak phase . Viewing the parameter space as a 1-D Brillouin zone, then for a two band Hamiltonian: $$H = h_x \sigma_x + h_y \sigma_y + h_z \sigma_z,$$ the Zak phase is half the solid angle of the winding path of the unit vector $$\hat{n} = (h_x, h_y, h_z)/ \sqrt{h_x^2+h_y^2+h_z^2}$$ on the Bloch sphere.
When the Hamiltonian has various symmetries, restrictions appear on the winding path, for example, when the Hamltonian has chiral symmetry, the winding path becomes a great circle and the result can assume the values of $0$ or $\pi$.
3. The values of the Zak phase are related to the $\mathbb{Z}_2$ invariants of the bands.
• @FraSchelle - Sorry for the late response. For a connection betwee the Zak phase and the existence of edge states, please see the following article arxiv.org/abs/1109.4608 by:Deiplace, Ulmo, Montambaux. For a connection bween the Zak phase and the $Z_2$ invariant, please see the following article arxiv.org/abs/1402.2434v1 by Grusdt, Abanin, Demler – David Bar Moshe Mar 8 '16 at 16:29