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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?

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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 $\pi$.

Several applications of the Zak phase were also found.

  1. The King-Smith-Vanderbilt formula relates the Zak phase to the polarization.
  2. The value of the Zak phase is related to the existence of edge states.
  3. The values of the Zak phase are related to the $\mathbb{Z}_2$ invariants of the bands.
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    $\begingroup$ Thanks for this answer, could you provide references for the different applications you provide about the Zak phase please ? In particular the relation between the Zak phase and edge states, or the Z2 invariant of the band is not clear to me. The KSV formula is demonstrated in journals.aps.org/prb/abstract/10.1103/PhysRevB.47.1651 . Thanks in advance for the other references. $\endgroup$ – FraSchelle Mar 4 '16 at 10:20
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    $\begingroup$ @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 $\endgroup$ – David Bar Moshe Mar 8 '16 at 16:29

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