# Periodic eigenfunctions for 2D Dirac operator

Consider the 2D Dirac operator

$$H = \begin{pmatrix} 0 & \partial_{\bar z} \\ \partial_z & 0 \end{pmatrix}$$

where $$\partial_z = \partial_x - i \partial_y$$ and $$\partial_{\bar z} = \partial_x + i \partial_y.$$

This implies by using the Bloch transform that there exist functions $$\psi_{\lambda}$$ such that

$$H \psi_{\lambda}= \lambda \psi_{\lambda}$$ and $$\psi_{\lambda}$$ are periodic on $$[0,1] \times [0,1]$$, i.e. $$\psi(x+1,y)=\psi(x,y)$$ and $$\psi(x,y+1)=\psi(x,y).$$

Can we find these functions explicitly?

To give an example where the situation is somewhat easier.

For the second derivative on $$\mathbb R$$ we have that

$$-\frac{d^2}{dx^2} e^{ikx} = k^2 e^{ikx}$$

where $$e^{ikx}$$ is periodic on $$[0,1]$$ if $$k \in 2\pi \mathbb{Z}.$$

$$\left( \begin{array}{c} 1 \\ \frac{k_x - ik_y }{\sqrt{k_x^2 + k_y^2 } } \end{array} \right) e^{i(k_x x + k_y y)} \ \ \ \mbox{with eigenvalue} \ \ \ i\sqrt{k_x^2 + k_y^2 }$$ and $$\left( \begin{array}{c} 1 \\ -\frac{k_x - ik_y }{\sqrt{k_x^2 + k_y^2 } } \end{array} \right) e^{i(k_x x + k_y y)} \ \ \ \mbox{with eigenvalue} \ \ \ -i\sqrt{k_x^2 + k_y^2 }$$ with $$k_x , k_y \in 2\pi \mathbb{Z}$$ for the desired periodicity.
The formulas of Michael Engelhardt's answer give the answer to the question. Let me add a short explanation how to achieve these formulas: The operator $$H$$ squares to the standard Laplacian on flat 2-space: $$H^2=\Delta$$. The periodic eigenfunctions of the Laplacian are well-known. Then, you have to fix an eigenvalue $$\lambda$$ of $$\Delta$$ and compute the eigenvectors of $$H$$ in the corresponding eigenspace $$Eig(\Delta,\lambda)$$. This is just a simple problem in linear algebra, and giv you the formulas in Michael Engelhardt's answer.