Podolsky quantum electrodynamics for strongly coupled Dirac fermions in (2+1)D

TL;DR

This work introduces PGQED, a dimensional reduction of GQED to (2+1)D with the Dirac current confined to a plane and a propagating gauge field, and uses Schwinger-Dyson equations to study dynamical mass generation for strongly coupled Dirac fermions. By applying rainbow-quenched and rainbow-unquenched truncations, the authors derive analytic expressions for the critical coupling $α_c(μ)$ and critical flavor number $N_c(μ)$ that depend on the Podolsky parameter $μ$ and UV cutoff $Λ$, and they uncover Miransky-type scaling of the generated mass in the strong-coupling regime. The Podolsky parameter acts to suppress mass generation (increasing $α_c$ and reducing $N_c$) but provides improved UV behavior relative to PQED; in the large-$μ$ limit the theory recovers PQED/QED3 results. The analysis is connected to graphene physics, with a suggested estimate of $μ$ (around 0.2 meV) for ultrarelativistic carriers in graphene, implying a small but potentially observable gap and illustrating how higher-derivative gauge dynamics can control electronic properties in two-dimensional materials.

Abstract

We investigate generalized quantum electrodynamics (GQED), a higher-derivative extension of QED in (3+1)D. We perform its dimensional reduction to (2+1)D by confining the Dirac current to a plane while allowing the gauge field to propagate out of the plane. The resulting model, which we call Pseudo Generalized QED (PGQED), is minimally coupled to massless Dirac fermions. In the strong-coupling regime, we show that a dynamical mass is generated through approximate solutions of the Schwinger-Dyson equations, leading to chiral symmetry breaking and modifications of the fermion dispersion relation. We derive an analytic critical coupling $α_{c}(μ)$ and flavors critical number $N_{c}(μ)$, dependent on the Podolsky parameter $μ$ and the ultraviolet cutoff $Λ$. These analytical results are found to be consistent with the numerical analysis. Finally, we connect our results to graphene, estimating a range for $μ$ in the ultrarelativistic limit and highlighting implications for two-dimensional materials.

Figures (9)

• Figure 1: The static potential of PGQED. The red line represents the PGQED potential, given by Eq. \ref{['Staticpot']}, and is plotted for $a=1$. The blue line represents the Coulomb potential of QED, obtained from Eq. \ref{['Staticpot']} in the limit $a\rightarrow \infty$. Note that the PGQED potential is finite at the origin.
• Figure 2: The Schwinger-Dyson equation for the gauge-field propagator in Eq. (\ref{['esdpp']}). Filled dots represent full propagators and vertex. The black dot corresponds to the full gauge-field propagator, the gray dot to the full fermion propagator, and the white dot to the vertex. The first term on the right-hand side is the bare gauge-field propagator, while the second term represents the gauge-field self-energy $\Pi^{\mu\nu}(p)$.
• Figure 3: The Schwinger-Dyson equation for the full electron propagator in Eq. (\ref{['SDfer']}). Filled dots represent full propagators and vertex. The black dot corresponds to the full gauge-field propagator, the gray dot to the full fermion propagator, and the white dot to the vertex. The first term on the right-hand side is the bare fermion propagator and the second term represents the fermion self-energy $\Xi(p)$.
• Figure 4: The $H(p,\mu)p^2\rightarrow H(\Lambda,\mu)p^2$ approximation within the large-momentum limit in Eq. (\ref{['EQDH']}). The blue line shows the exact expression, where we plot $p^2$ times Eq. \ref{['funcH']} for $\mu=10$. The red line corresponds to the large-momentum approximation, plotting $p^2$ times Eq. \ref{['funcH']} for $\mu=10$ and $\Lambda=10$. This approximation is valid as long as $\Lambda>p$ with a finite $\Lambda$.
• Figure 5: The critical coupling constant of PGQED in Eq. (\ref{['alcqr']}). The blue line shows Eq. \ref{['alcqr']} with $\mu\rightarrow\infty$, which corresponds to the critical coupling constant in PQED Alves2013. The red line shows Eq. \ref{['alcqr']} for $\Lambda=10$. We observe that as $\mu$ decreases, $\alpha_c$ increases which makes dynamical mass generation even harder to be realized because it needs $\alpha>\alpha_c(\mu)$.