Weyl fermions

Weyl Fermions ¹¶

The Dirac equation in d spatial dimension and effective speed of light c = 1 is \(\((i\gamma^\mu \partial_\mu -m) = 0\)\) where, \(\mu = 0,1, \cdots d\) label time and space dimensions, and the d+1 gamma matrices satisfy the anticommutation relation \(\{\gamma^\mu,\gamma^\nu\} =0\) for \(\mu \ne \nu\) and \({(\gamma^0)}^2 = {(\gamma^i)}^2 = \mathcal{I}^{2 \times 2}\), where \(i = 1, \cdots d\) and \(\mathcal{I}^{2 \times 2}\) is the \(2 \times 2\) matrix.

When the mass term in the Dirac equation vanishes, two independent solutions appear. They are chiral fermions with opposite handedness, called Weyl fermions. Although they have not been discovered as a type of fundamental particle within the Standard Model, quasiparticles with these properties were discovered about five years ago in condensed matter systems called Weyl semimetals.

Weyl noticed that this equation can be further simplified in certain cases in odd spatial dimensions. For simplicity, consider d =1. Then one needs only two anticommuting matrices, e.g., the \(2 \times 2\) Pauli matrices e.g.,\(\gamma^0 = \sigma_z\) and \(\gamma_1 = i \sigma_y\). Therefore, the Dirac equation in 1 + 1 dimension involves a two component spinor and can be written as \(\(i\partial_t \psi = (\gamma^0\gamma^1p + m\gamma^0)\)\), where \(p = -i\partial_x\). If one were describing a massless particle m = 0, this equation can be further simplified by simply picking eigenstates of the Hermitian matrix \(\gamma_5 = \gamma^0 \gamma^1 = \sigma^x\) if \(\gamma_5\psi_{\pm} = \pm\psi_\pm\). One then has the 1D Weyl equation \(\(i\partial_t\psi_\pm = \pm p\psi_\pm\)\) The resulting dispersion is simply \(E_\pm(p) =\pm p\) which denotes a right (left) moving particle, which are termed chiral or Weyl fermions.

The chirality of the Weyl fermion comes from the fact that the direction of its pseudospin is always locked to the direction of the translational motion and is either parallel (positive chirality) or antiparallel (negative chirality) to it. The chiral electronic structure leads to particular charge dynamics: the Chiral Anomaly and Chiral magnetic effect