  We consider conformal immersions $f: T^2\rightarrow \mathbb{R}^3$ with the
property that $H^2 f^*g_{\mathbb{R}^3}$ is a flat metric. These so called Dirac
tori have the property that its Willmore energy is uniformly distributed over
the surface and can be obtained using spin transformations of the plane by
eigenvectors of the standard Dirac operator for a fixed eigenvalue. We classify
Dirac tori and determine the conformal classes realized by them. We want to
note that the spinors of Dirac tori satisfies the same system of PDE's as the
differential of Hamiltonian stationary Lagrangian tori in $\mathbb{R}^4$. These
were classified in [5] .
