Periodic eigenfunctions for 2D Dirac operator

$$ \left( \begin{array}{c} 1 \\ \frac{k_x - ik_y }{\sqrt{k_x^2 + k_y^2 } } \end{array} \right) e^{i(k_x x + k_y y)} \ \ \ \mbox{with eigenvalue} \ \ \ i\sqrt{k_x^2 + k_y^2 } $$ 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.