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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 $k \in 2\pi \mathbb{Z}.$

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2 Answers 2

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

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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.

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