Dirac semimetal are 3D versions of 2D graphene. In case of Graphene the Dirac point are only in the corner of hexagonal Briloune Zone which are time and space reversely protected. In Dirac semimetal, these Dirac points are now inside the Brilouine Zone along the symmetry axis.
Grphene Dirac cone
Dirac semimetal Dirac cone
This happen by considering the point group symmetry along with Time reversal and space inversion symmetry..PHYSICAL REVIEW B 85, 195320 a material prediction in Na3Bi in the year 2012.
In Na3Bi there is band crossing because of spin orbit interaction at K_D and there is other one also at -K_D. Now such crossing are protected against gap formation by point group symmetry as they they different representation they can"t never hybridize.
Na3Bi has C3 point group symmetry (http://science.sciencemag.org/content/343/6173/864/tab-pdf)
Cd3As2 has C4 symmetry (https://www.nature.com/articles/nmat3990
)
As already discussed if we broke either time reversal (by applying magnetic field or having a spontaneous magnetization ) or space inversion the we will get Wyel semimetal. As theoriticaly predicted in 2011 by Vishwanath
Weyl found an important conclusion of Dirac equation:
1>Start with Durac equtaion and set the mass zero
2?Do not include any electromagnetic interaction
Then we have a mass less Dirac Fermion with two different population of opposite chirality and they never mixed up. When there is mass term then this two chirality will mixed up.
In 2D this equivalent to parity. So let's talk about chirality in more details:
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