**2.1. The parity operator**

Let us consider the parity operator in (3+1)QED [17]

$$
\widehat{P} = i\gamma\_4 \widehat{\Lambda}\_{\mathbf{n}}.\tag{1}
$$

Here, *iγ*<sup>4</sup> is the inversion operator and

$$
\widehat{\Lambda}\_{\mathbf{n}} = e^{-i\frac{\pi}{2}\Sigma\mathbf{n}} = -i\Sigma\mathbf{n}
$$

is the operator of rotation by *π* about an **n** axis perpendicular to the graphene plane. In standard representation,

$$
\Sigma = \begin{pmatrix} \sigma \ 0 \\ 0 \end{pmatrix}\_{\prime}
$$

where σ denotes Pauli matrices, and

$$
\gamma\_4 \equiv \mathcal{J} = \begin{pmatrix} I & 0 \\ 0 & -I \end{pmatrix}
$$

where *I* is the 2×2 unit matrix. It is clear that operator (1) is analogous to the parity operator *<sup>i</sup>γ*5*<sup>n</sup>* in QED, where *<sup>n</sup>* <sup>=</sup> <sup>γ</sup>**n**,

$$
\gamma = \begin{pmatrix} \begin{smallmatrix} 0 & \sigma \\ -\sigma & 0 \end{smallmatrix} \end{pmatrix}, \quad \gamma\_5 = \gamma\_1 \gamma\_2 \gamma\_3 \gamma\_4 = i \begin{pmatrix} \begin{smallmatrix} 0 \\ I & 0 \end{smallmatrix} \end{pmatrix}.
$$

The eigen functions of this operator describe electron polarization states [18].

Charge carrier states in graphene can be described in terms of helicity defined as the eigenvalue of the operator *h* = σ · **p**/(2|**p**|). The projection of pseudospin on the direction of quasimomentum *p* indicates the valley in the Brillouin zone where electrons or holes belong (*K* or *K* point in Fig. 2). Positive helicity corresponds to electrons and holes with wavevectors near the *K* and *K* points, respectively; negative helicity corresponds to electrons and holes with wavevectors near the *K* and *K* points, respectively [19].

Massless states with opposite helicities are decoupled [20]. In addition, charge carriers have chiral symmetry (helicity is conserved), and parity can be defined for both massless and massive carriers1. In other words, a higher symmetry of massless charge carriers implies the existence of an additional quantum number: helicity. Whereas parity distinguishes only between the valleys where carrier states belong (*λ* = +1 and −1 for states close to the *K* and *K* points, respectively), helicity differs between a particle (electron) and an antiparticle (hole). However, chiral symmetry is broken for massive charge carriers. So, helicity is not a good quantum number any longer. Carrier states in a planar heterostructure combining gapless and gapped graphene should be characterized by parity.

Recall that the Dirac equation describing massless carriers in graphene in terms of 4×4 matrices is derived by assuming that they are spinless and have two valley degrees of freedom [10]. When analysis is restricted to charge carriers in one valley, the Dirac equation can be reduced to a 2×2 matrix representation by Weyl's equation for a massless fermion analogous to neutrino in two Euclidean dimensions [22]. The carrier energy spectrum with a pseudospin splitting in a planar heterostructure combining gapless and gapped graphene cannot be correctly analyzed in the 2×2 representation. For similar reasons, the representation of the Dirac algebra in terms of 2×2 matrices is not sufficient for describing the chiral symmetry breaking in (2+1)QED [23].

Using the two-dimensional 4×4 Dirac equation to describe charge carriers in a graphene-based nanostructure, we can study pseudospin effects following an approach to narrow-gap semiconductor heterostructures based on the Dirac model [24]. This makes methods developed for solving problems in the spintronics of narrow-gap semiconductor heterostructures applicable to graphene-based ones [25–33].
