Induced Maxwell-Chern-Simons Effective Action in Very Special Relativity

In this paper, we study the one-loop induced photon's effective action in the very special relativity electrodynamics in $(2+1)$ spacetime (VSR-QED$_{3}$). Due to the presence of new nonlocal couplings resulting from the VSR gauge symmetry, we have additional graphs contributing to the $\langle AA\rangle$ and $\langle AAA \rangle$ amplitudes. From these contributions, we discuss the VSR generalization of the Abelian Maxwell-Chern-Simons Lagrangian, consisting in the dynamical part and the Chern-Simons-like self-couplings, respectively. We use the VSR-Chern-Simons electrodynamics to discuss some non-Ohmic behavior on topological materials, in particular VSR effects on Hall's conductivity. In the dynamical part of the effective action, we observe the presence of a UV/IR mixing, due to the entanglement of the VSR nonlocal effects to the quantum higher-derivative terms. Furthermore, in the self-coupling aspect, we verify the validity of the Furry's theorem in the VSR-QED$_{3}$ explicitly.