# Why aren't the eigenvectors of a tight-binding Hamiltonian periodic?

I try to calculate the Berry connection for a simple graphene model and stumbled across the following question. Suppose I have a tight binding Hamiltonian (further details here or here): $$H = \begin{pmatrix} 0 & f \\ f^* & 0 \end{pmatrix}, \qquad f = \exp(ik_xa) + 2\exp\left(-ik_x \frac{a}{2}\right) \cos\left(k_y \frac{\sqrt{3}}{2}a\right) \\$$ with the k-dependent eigenvalues and eigenvectors $$E_{c/v} = \pm |f(k)|, \qquad |c/v, k\rangle = \begin{pmatrix} 1 \\ \pm e^{-i\alpha(k)} \end{pmatrix}, \qquad \alpha(k) = \arg(f(k)).$$ The BZ is oriented in a way, in which $$\Gamma = (0, 0)$$ and $$M=(2\pi/3a, 0)$$. If I evaluate the energies for - for instance - the valence band at the $$M$$-point on the left and right edge of the BZ I, as one would expect, find $$E_v((-2\pi/3a, 0)) = E_v((2\pi/3a, 0))$$. Comparing the eigenvectors however, leads up to a different result, i.e. $$|v, (-2\pi/3a, 0)\rangle = \begin{pmatrix} -1 \\ -\frac{1+\sqrt{3}i}{2} \end{pmatrix}, \qquad |v, (2\pi/3a, 0)\rangle = \begin{pmatrix} -1 \\ -\frac{1-\sqrt{3}i}{2} \end{pmatrix}$$ I thought those vectors should be the same (up to a phase transformation), since we look at symmetric points. What am I getting wrong here?

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To clear things up, I added a picture showing the valence band energy, the BZ borders and the imaginary and real parts of the vector components along path marked in red. As one can clearly see, the energy is indeed periodic but the vector components are neither periodic nor continuous and there is no phase function too make them at least continuous.

• @march It depends on the definition of $a$ (see ref). The $k_y$ - direction is less important, since it corresponds to the $K$-points of the BZ. I also checked numerically with my own code and pythTB that the eigenvectors for different $M$ points are different. Commented Dec 13, 2021 at 18:11

I found a publication that explained the problem. When constructing the wavefunctions, one combines orbitals from both sublattices (A and B) of graphene. Two standard options are a single phase factor for both sublattices $$\Psi_I^k(r) = \frac{1}{\sqrt{N}}\sum_j e^{ik\cdot R_j} \left(c_I^A(k)\phi(r-R_j^A) + c_I^B(k)\phi(r-R_j^B)\right),$$ where $$R_j$$ is a lattice vector or two separate phase factors $$\Psi_{II}^k(r) = \frac{1}{\sqrt{N}}\sum_j \left(e^{ik\cdot R^A_j}c_{II}^A(k)\phi(r-R_j^A) + e^{ik\cdot R^B_j}c_{II}^B(k)\phi(r-R_j^B)\right).$$ The system I presented in the question corresponds to the second definition in which the functions $$c^i_{j}(k)$$ are not periodic in the BZ.