We present a self-dual parity-invariant $U(1)\times U(1)$ Maxwell-Chern-Simons scalar ${\text{QED}}_3$. We show that the energy functional admits a Bogomol'nyi-type lower bound, whose saturation gives rise to first order self-duality equations. We perform a detailed analysis of this system, discussing its main features and exhibiting explicit numerical solutions corresponding to finite-energy topological vortices and non-topological solitons. The mixed Chern-Simons term plays an important role here, ensuring the main properties of the model and suggesting possible applications in condensed matter.

中文翻译：

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奇偶不变场景中的自对偶 Maxwell-Chern-Simons 孤子

我们提出了一个自对偶奇偶不变量$ü(1)\times ü(1)$Maxwell-Chern-Simons 标量${\text{量子点}}_3$. 我们证明能量泛函具有 Bogomol'nyi 型下界，其饱和度产生一阶自对偶方程。我们对该系统进行了详细的分析，讨论了它的主要特征，并展示了与有限能量拓扑涡旋和非拓扑孤子相对应的显式数值解。混合 Chern-Simons 项在这里起着重要作用，确保模型的主要特性并建议在凝聚态物质中的可能应用。