Unlike Z2 class of topological insulator where the states are protected by TR and SI symmetry. InTCI it is the crystal symmetry that protect it. Time-reversal symmetry protects strong topological insulators of the Z2 class, which possess an odd number of metallic surface states with dispersion of a Dirac cone. Topological crystalline insulators are merely protected by individual crystal symmetries and exist for an even number of Dirac cones. Here we mirror chern number.
SnTe is an example where the band inversion is protected by mirror symmetry but it is not the case for PbTe
In thus case te surface states Dirac cone neither is at corner of Brilouin Zone (Z2 TI) or in the symetry axis (Dirac semimetal) rather it is situated on mirror inversion point. inset of bottom of figure.
Unlike Graphene these Dirac point can"t annihilate. So what is the difference between TI and TCI ?
In TI there are odd no of Dirac cone situated on time reversal point in momentum space but for TCI there are even no of Dirac cone which are situated any where except the time reversal point.
Now one interesting point is if we break the mirror inversion symmetry then a gap will appear in Dirac cone which will lead to appearance of mass. This symmetry can be broken by structural distortion upon lowering the temperature. There is a report here
http://science.sciencemag.org/content/341/6153/1496
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Prediction by a toy model (2010) https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.106.106802
SnTe (2012): https://www.nature.com/articles/nphys2442
Observation of ferroelectricity and proposal of ferroelectric tunneling random access memory.!!
A complete review by Liang Fu;
This paper demonstrate between Z2 to TCI: https://www.nature.com/articles/s41467-017-01204-0
Further Idea:
Hourglass fermions :https://www.nature.com/articles/nature17410
Non symorphic TI: https://journals.aps.org/prb/references/10.1103/PhysRevB.96.165143
One point to remember upto now we have not consider any interaction in TCI !!
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