Abstract
We derive two versions of an effective model to describe dynamical effects of the Yukawa interaction among Dirac electrons in the plane. Such short-range interaction is obtained by introducing a mass term for the intermediate particle, which may be either scalar or an abelian gauge field, both of them in ($3+1$) dimensions. Thereafter, we consider that the fermionic matter field propagates only in ($2+1$) dimensions, whereas the bosonic field is free to propagate out of the plane. Within these assumptions, we apply a mechanism for dimensional reduction, which yields an effective model in ($2+1$) dimensions. In particular, for the gauge-field case, we use the Stueckelberg mechanism in order to preserve gauge invariance. We refer to this version as nonlocal-Proca quantum electrodynamics (NPQED). For both scalar and gauge cases, the effective models reproduce the usual Yukawa interaction in the static limit. By means of perturbation theory at one loop, we calculate the mass renormalization of the Dirac field. Our model is a generalization of Pseudo quantum electrodynamics (PQED), which is a gauge-field model that provides a Coulomb interaction for two-dimensional electrons. Possibilities of application to Fermi-Bose mixtures in mixed dimensions, using cold atoms, are briefly discussed.
Highlights
In the last decades, the interest of studying planar theories has increased in theoretical physics, mainly because of the discovery of new quantum effects, such as high-Tc superconductivity, quantum Hall effect, and topological phase transitions [1]
For the gauge-field case, we use the Stueckelberg mechanism in order to preserve gauge invariance. We refer to this version as nonlocal-Proca quantum electrodynamics (NPQED)
Our model is a generalization of Pseudo quantum electrodynamics (PQED), which is a gauge-field model that provides a Coulomb interaction for two-dimensional electrons
Summary
The interest of studying planar theories has increased in theoretical physics, mainly because of the discovery of new quantum effects, such as high-Tc superconductivity, quantum Hall effect, and topological phase transitions [1]. [22] for applications in the realm of topological insulators All of these works rely on the fact that PQED generates a long-range interaction in the static limit, namely, the Coulomb potential VCðrÞ ∝ 1=r. Within the quantum-fieldtheory interpretation, we may claim that the mediating field has a mass term Motivated by this well-known result, we shall use the paradigm of including a mass term, for the intermediate particle, in order to generate a short-range interaction, i.e., the Yukawa potential in the plane. Since a naive addition of a mass for the gauge field would break gauge invariance, we consider the well known Stueckelberg action This model is a generalized version of Proca quantum electrodynamics, on which we perform the dimensional reduction. We include one Appendix, where we present the details about the electron-self energy as well as the corrected gauge-field propagator at one loop in perturbation theory
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