Functional integral method in quantum field theory of plasmons in graphene

Nguyen Duc Duoc Phan and Van Hau Tran

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Keywords: nano


In the present work we apply the functional integral method to the study of quantum field theory of collective excitations of spinless Dirac fermion in graphene at vanishing absolute temperature and at Fermi level ${{E}_{{\rm F}}}=0$. After introducing the Hermitian scalar field $\varphi (x)$ describing these collective excitations we establish the expression of the functional integral ${{Z}^{\varphi }}$ containing a functional series $I[\varphi ]$. The explicit expressions of several terms of this functional series were derived. Then we consider the functional series $I[\varphi ]$ in second order approximation and denote ${{I}_{0}}[\varphi ]$ the corresponding approximate expression of $I[\varphi ]$. We shall demonstrate that in this approximation the scalar field $\varphi (x)$ can be devided into two parts: a background field ${{\varphi }_{0}}(x)$ corresponding to the extremum of ${{I}_{0}}[\varphi ]$ and another scalar field $\xi (x)$ describing the fluctuation of $\varphi (x)$around the background ${{\varphi }_{0}}(x)$. We call $\xi (x)$ the fluctuation field. Then we establish the relationship between this fluctuation field $\xi (x)$ and the quantum field of plasmons in graphene. Considering some range of values of frequency (energy) and wave vector (momentum) of plasmons, when the analytical calculations can be performed, we derived the differential equation for the quantum field of graphene plasmons. From this field equation we establish the relation between frequency and wave vector of plasmons in the long wavelength limit

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