# The strange behave of wave function of the hexagonal lattice. Not single-valued, strange at K and K' Points

I want to know a wave function should be periodic or not in the Brillouin Zone.

I calculated the eigenvalues and wave function of the hexagonal lattice by the tight-binding approach. The wave function of one electron in the hexagonal lattice is given as the linear combination of the Bloch functions $\Phi_A$ and $\Phi_B$, as

$$\Psi^{\lambda}({\bf k},{\bf r})=C_A^{\lambda}({\bf k})\Phi_A({\bf k},{\bf r})+C_B^{\lambda}({\bf k})\Phi_B({\bf k},{\bf r}) ,$$

where $\lambda$ is $c$ or $v$, i.e. an index running over the conduction and valence bands. The Hamiltonian $H$ is given as $$\begin{pmatrix} 0 & tf({\bf k}) \\ tf({\bf k})^* & 0 \\ \end{pmatrix},$$ where $f({\bf k})$ is the sum of the phases of the nearest hopping and defined as $$f({\bf k})=e^{i{\bf k}\cdot {\bf R}_1}+e^{i{\bf k}\cdot {\bf R}_2}+e^{i{\bf k}\cdot {\bf R}_3},$$ ${\bf R}_1,{\bf R}_2,{\bf R}_3$ are the paths of the nearest hopping, and $a$ is lattice constant.

I solve the equation $H{\bf C}^{\lambda}=E^{\lambda}{\bf C}^{\lambda}$. $E({\bf k})^{\lambda}$ is the eigenvalue and ${\bf C(k)}^{\lambda}$ is called pseudo spin and defined as $$\begin{pmatrix} C_A^{\lambda}({\bf k}) \\ C_B^{\lambda}({\bf k}) \\ \end{pmatrix},$$ the solutions of which are given by $${\bf C}^c = {\sqrt \frac{1}{2w({\bf k})}}\begin{pmatrix} \sqrt{f({\bf k})} \\ -\sqrt{f({\bf k})^*} \\ \end{pmatrix},\quad {\bf C}^v = {\sqrt \frac{1}{2w({\bf k})}}\begin{pmatrix} \sqrt{f({\bf k})} \\ \sqrt{f({\bf k})^*} \\ \end{pmatrix}$$ in terms of $w({\bf k})=|f({\bf k})|$.

Actually, $f({\bf k})$ is not periodic in k-space. That is, $f({\bf k}) \neq f({\bf k}+p{\bf G}_1+q{\bf G}_2)$, where $p$ and $q$ are integers and ${\bf G}_1$ and ${\bf G_2}$ are reciprocal lattice vectors given by \begin{align} {\bf G}_1&=\left(\frac{2\sqrt{3}\pi}{3a},\frac{2\pi}{a}\right),\\ {\bf G}_2&=\left(\frac{2\sqrt{3}\pi}{3a},-\frac{2\pi}{a}\right). \end{align}

The wave function of this system is not periodic because $f({\bf k})$ is not periodic. So the wavefunction is not single-valued function.

My queastions:

1. Is it ok? I think the wave function should be a single-valued function.

2. Can we choose the phase of the wavefunction so that the wave function is a single-valued function? For example, $${\bf C}^c = {\sqrt \frac{1}{2}}\begin{pmatrix} 1 \\ -{\sqrt \frac{f({\bf k})}{f({\bf k})^*}} \\ \end{pmatrix},\quad {\bf C}^v = {\sqrt \frac{1}{2}}\begin{pmatrix} {\sqrt \frac{f({\bf k})}{f({\bf k})^*}} \\ 1 \\ \end{pmatrix}$$ can be eigenfuncitons of the Hamiltonian. But even in this case, the wavefunction is not single-valued function.

3. Why is the wavefunction indeterminate at the K and K' points? At the K and K' points, the pseudo spins can not be defined because $f({\bf K})=f({\bf K'})=0$. The pseudo spins include $\frac{0}{0}$. How can we avoid this strange behaviour?

4. How about other systems? Such a funny phenomena occur in only hexagonal lattice? I mean ─ is the hexagonal lattice special? or does this behaviour occur elsewhere?

• Why do you say that $f$ is not periodic? As far as I can see, with your notation, $f({\bf k}+p{\bf G}_1+q{\bf G}_2)=e^{i(p+q)2\pi/3}f({\bf k})$, which is almost as good as periodic. – Emilio Pisanty Jun 23 '17 at 10:38
• Thank you for your comment. I agree. It is true $f({\bf k})$ is periodic. But the period is three times of the Brillouin Zone. For example, $f({\bf k+G_1+2G_2})=f({\bf k})$ but $f({\bf k+G_1})\neq f({\bf k})$. I think the period of $f({\bf k})$ should be Brillouin Zone if the wave function should be a single-valued function. – Sakurai.JJ Jun 23 '17 at 11:01
• Look at the paper devoted exactly to this issue: iopscience.iop.org/article/10.1088/1367-2630/11/9/095003 – Alexey Sokolik Jun 23 '17 at 15:15
• To Alexey Sokolik. I have read the paper. But I don't think the paper answer my questions. The paper just says "the expectation values don't depend on the phase of the wave function". – Sakurai.JJ Jun 25 '17 at 5:35