The anomalous quantum Hall conductivity in the 2 + 1 D topological insulators in the absence of interactions may be expressed as the topological invariant composed of the two-point Green function. For the noninteracting system this expression is the alternative way to represent the TKNN invariant. It is widely believed that in the presence of interactions the Hall conductivity is given by the same expression, in which the noninteracting two-point Green function is substituted by the complete two-point Green function with the interactions taken into account. However, the proof of this statement has not been given so far. In the present paper we give such a proof in the framework of the particular tight-binding models of the 2 + 1 D topological insulator. Besides, we extend our consideration to the 3 + 1 D Weyl semimetals. It was known previously that with the interactions neglected the Hall conductivity in those systems is expressed through the two-point Green function in the way similar to that of the 2 + 1 D topological insulators. Again, the influence of interactions on this expression has not been investigated previously. We consider this problem within the framework of the particular 3 + 1 D model of Weyl semimetal in the presence of the contact four-fermion interactions and Coulomb interactions. We prove (up to the one-loop approximation), that the Hall conductivity is given by the same expression as in the noninteracting case, in which the noninteracting Green function is substituted by the complete two-point Green function with the interactions included. Basing on the obtained expressions we discuss the topological phase transitions accompanied by the change of Hall conductivity.

在没有相互作用的情况下，2+1D 拓扑绝缘体中的反常量子霍尔电导率可以表示为由两点格林函数组成的拓扑不变量。对于非交互系统，此表达式是表示 TKNN 不变量的另一种方式。人们普遍认为，在存在相互作用的情况下，霍尔电导率由相同的表达式给出，其中非相互作用的两点格林函数被完整的两点格林函数替代，并考虑了相互作用。但是，目前还没有给出这一说法的证据。在本文中，我们在 2 + 1 D 拓扑绝缘体的特定紧束缚模型的框架中给出了这样的证明。此外，我们将考虑扩展到 3 + 1 D Weyl 半金属。先前已知，在忽略相互作用的情况下，这些系统中的霍尔电导率通过两点格林函数以类似于 2 + 1 D 拓扑绝缘体的方式表示。同样，以前没有研究过相互作用对这种表达的影响。我们在存在接触四费米子相互作用和库仑相互作用的情况下，在 Weyl 半金属的特定 3 + 1 D 模型的框架内考虑这个问题。我们证明（直到单环近似），霍尔电导率由与非相互作用情况相同的表达式给出，其中非相互作用格林函数被包含相互作用的完整两点格林函数代替。