The Effective Potential of Scalar Pseudo-Quantum Electrodynamics in (2 + 1)D

Abstract

The description of the electron–electron interactions in two-dimensional materials has a dimensional mismatch, where electrons live in (2 + 1)D while photons propagate in (3 + 1)D. In order to define an action in (2 + 1)D, one may perform a dimensional reduction of quantum electrodynamics in (3 + 1)D (QED4) into pseudo-quantum electrodynamics (PQED). The main difference between this model and QED4 is the presence of a pseudo-differential operator in the Maxwell term. However, besides the Coulomb repulsion, electrons in a material are subjected to several microscopic interactions, which are inherent in a many-body system. These are expected to reduce the range of the Coulomb potential, leading to a short-range interaction. Here, we consider the coupling to a scalar field in PQED for explaining such a mechanism, which resembles the spontaneous symmetry breaking (SSB) in Abelian gauge theories. In order to do so, we consider two cases: (i) by coupling the quantum electrodynamics to a Higgs field in (3 + 1)D and, thereafter, performing the dimensional reduction; and (ii) by coupling a Higgs field to the gauge field in PQED and, subsequently, calculating its effective potential. In case (i), we obtain a model describing electrons interacting through the Yukawa potential and, in case (ii), we show that SSB does not occur at one-loop approximation. The relevance of the model for describing electronic interactions in two-dimensional materials is also addressed.