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Pauli equations and non-commutative position operators in 2D Dirac-like semiconductors in view of second quantization

The Pauli equations describing electron (hole) dynamics in 2D Dirac-like intrinsic semiconductors in external (impurity) scalar potential and for inhomogeneous lattice distortions are obtained within second quantization approach. We show that the modifying external perturbation terms in formulated no-pair equations are in general non-local and demonstrates singular behavior in gapless situation where they do not depend on semiconductor parameters. It is shown that lattice distortion perturbation can cause confinement of both electrons and holes in the same spatial region. The proposed approach is verified by comparison with well-established results in low energy limit. The consideration of position operator in second quantization approach allows elucidating the physical meaning of spin-orbit-like and Darwin-like terms.