For an overview of the relevant details, consult the contribution by McCann and Fal'ko in [1] (Chapter 10). The group-theoretical considerations as encountered in [1] are discussed in detail in [2]. For a comprehensive discussion of the concept of isospin, consult [3]. Briefly, the set of operators (or their matrix representations) that are closed under commutation form an algebra named Lie algebra [4]. The operators associated with orbital and spin angular momenta form Lie algebras. Both sets of operators {Σx, Σy, Σz} and {Λx, Λy, Λz} as encountered in the description of the electronic Hamiltonian for (non-magnetically) disordered graphene form Lie algebras, as can be immediately inferred from the explicit expressions given before Eq. (10.5) in [1] (p. 331). Incidentally, in [1] (as well as in the relevant references herein) one can find the rationale for using the above two sets of isospin operators for describing the electronic Hamiltonian of a disordered graphene sheet [5].

[1] H Aoki, and MS Dresselhaus, editors, Physics of Graphene (Springer, Heidelberg, 2014).

[2] DM Basko, Phys. Rev. B 78, 125418 (2008).

[3a] HJ Lipkin, Lie Groups for Pedestrians (Dover, New York, 2002).

For a more advanced treatment, consult:

[3b] F Iachello, Lie Algebras and Applications, 2nd edition (Springer, Berlin, 2015).

[4] For {Aα | α} forming a Lie algebra, one has the closure relation

[Aα,Aβ] = Σγ Cγα,β Aγ,

where {Cγα,β | α, β, γ} are constants. Since [A,B] = - [B,A], one further has

Cγα,β = - Cγβ,α.

[5] This in order to make the time-reversal invariance of the electronic Hamiltonian corresponding to a graphene sheet subject to the potentials of non-magnetic impurities explicit.

Dear Mukesh,

The lattice of the graphene (honeycomb symmetry which is not a Bravais lattice) has two Bravais lattices and therefore two distinct Dirac points in the Brillouin zone. This allows to make an analogy with the two states of a spin, up-down, and to find a SU(2) representation called pseudo-spin. That is to say, employing the Pauli matrices to work with then in a tight binding model as if they were properly spins. On the other hand the six electrons of the carban have also proper spins. The four hybrideced 2s2 2p2 electrons therefore can have both spin and pseudospin in its physical properties,i.e. with a SU(4) general symmetry, at difference of what happens with most of the non 2-D structures.

I hope that this can help to understand this subtle issue.

Daniel Baldomir Sir,

I would like to add something, i hope that this will work.

In the direct lattice(position space) of graphene, two non-equivalent carbon atom positions (corresponding to sublattice A and sublattice B respectively) indicate the analogy of pseudo-spin but on the other hand if we go for the reciprocal lattice (k-space), the two non- equivalent conical self crossing points or dirac points represent the Isospin =2. In addition to that the hybridized electrons in carbon intrinsically have real spins. Thereby getting the idea of degeneracy of electrons in graphene that is 4.

Dear Mukesh,

The degeneracy is only two, the position or kspaces cannot add their degeneracies. On the other hand the four electrons (in fact six) only can be considered for given a honeycomb structure on 2-D, after that the bands are what can produce the Dirac points, nothing more.

Dear Mukesh,

It is true that you are right that you have four degeneracies of the graphene system assuming the four symmetries of the lattice (not of the electrons)

- Translations in each Bravais sublattice A to A and B to B.

- Mirror reflection between A and B

- A rotation symmetry of sixty degrees ( C pi/3)

But that is independent of what I have been speaking about the isospin and spin or the electronic degeneration. This is only due to the lattice symmetries. Is this what you were thinking about?