It is commonly stated that the low energy theory of some non-relativistic fermion band structure with a linear dispersion $E = vk$ implies the

  1. low energy effective theory is a relativistic fermion theory.

  2. there is an emergent Lorentz symmetry at or near the node

Of course $E = vk$ is necessary condition, but it does not seem to be sufficient to show 1 and 2. Also the Lorentz spinor degrees of freedom is unclear.

My questions

  • why 1 and 2 are true? What conditions guarantee those? (Sketch a physics proof?)

  • How to determine which relativistic fermions they have at low energy: Dirac, Majorana, Weyl or Majorana-Weyl spinors? as Lorentz spinors?

  • why are they Lorentz spinors? (they can be approximate spatial rotational symmetries, but not the Lorentz spacetime boost + rotational symmetries ?)

enter image description here

Figure from https://dx.doi.org/10.1103/PhysRevB.86.155439 https://arxiv.org/abs/1210.7037