The basic materials of Berry’s phase and chiral anomalies are presented to appreciate the phenomena related to those notions recently discussed in the literature. As for Berry’s phase, a general survey of the subject including the anomalous Hall effect is presented using both Lagrangian and Hamiltonian formalisms. The canonical Hamiltonian formalism of the Born–Oppenheimer approximation, when applied to the anomalous Hall effect, can incorporate the gauge symmetry of Berry’s connection but unable to incorporate the completely independent gauge symmetry of the electromagnetic vector potential simultaneously. Thus the Nernst effect is not realized in the canonical formalism. Transformed to the Lagrangian formalism with a time-derivative term allowed, the Born–Oppenheimer approximation can incorporate the electromagnetic vector potential simultaneously with Berry’s connection, but the consistent canonical property is lost and thus becomes classical. The Lagrangian formalism can thus incorporate both gauge symmetries simultaneously but spoils the basic quantum symmetries, and thus results in classical anomalous Poisson brackets and the classical Nernst effect as in the conventional formalism. These properties are taken as the bases of the applications of Berry’s phase to the anomalous Hall effect in the present review.

As for chiral anomalies, we present basic materials by the path integral formulation with an emphasis on fermions on the lattice. A chiral fermion defined by ${\gamma }_5$ on the lattice does not contain the chiral anomaly for the non-vanishing lattice spacing $a\ne 0$. Each species doubler separately does not contain a well-defined chiral anomaly either, since each species doubler defined in a part of the Brillouin zone is not a local field for $a\ne 0$. The idea of a spectral flow on the lattice does not lead to an anomaly for each species doubler separately but rather to a pair production in a general sense. We also mention that a specific construction called the Ginsparg–Wilson fermion, which is free of species doublers, may practically be useful in the theoretical analysis of an Abelian massless Dirac fermion in condensed matter physics.

We discuss a limited number of representative applications of Berry’s phase and chiral anomalies in nuclear physics and related fields to illustrate the use of these two basic notions presented in this article.

介绍了 Berry 相和手性异常的基本材料，以了解与文献中最近讨论的那些概念相关的现象。至于 Berry 相，使用拉格朗日和哈密顿形式主义对包括异常霍尔效应在内的主题进行了一般调查。Born-Oppenheimer 近似的规范哈密顿形式在应用于反常霍尔效应时，可以包含 Berry 连接的规范对称性，但无法同时包含电磁矢量势的完全独立的规范对称性。因此，能斯特效应在规范形式主义中没有实现。转换为允许时间导数项的拉格朗日形式，Born-Oppenheimer 近似可以将电磁矢势与 Berry 联系同时合并，但失去了一致的规范性质，因此成为经典的。因此，拉格朗日形式主义可以同时包含两个规范对称性，但破坏了基本的量子对称性，因此导致经典反常泊松括号和传统形式主义中的经典能斯特效应。这些性质被视为本综述中贝瑞相在反常霍尔效应中应用的基础。从而导致经典的反常泊松括号和传统形式主义中的经典能斯特效应。这些性质被视为本综述中贝瑞相在反常霍尔效应中应用的基础。从而导致经典的反常泊松括号和传统形式主义中的经典能斯特效应。这些性质被视为本综述中贝瑞相在反常霍尔效应中应用的基础。

至于手性异常，我们通过路径积分公式呈现基本材料，重点是晶格上的费米子。手性费米子定义为${\gamma }_{\mathrm{5个}}$晶格上不包含非零晶格间距的手性异常$\mathrm{一个}\ne 0$. 每个物种倍增器单独也不包含明确定义的手性异常，因为在布里渊区的一部分中定义的每个物种倍增器不是局部场$\mathrm{一个}\ne 0$. 晶格上的光谱流的想法不会分别导致每个物种加倍器的异常，而是导致一般意义上的成对产生。我们还提到了一种称为 Ginsparg-Wilson 费米子的特定结构，它没有物种倍增器，实际上可能对凝聚态物理学中的阿贝尔无质量狄拉克费米子的理论分析很有用。

我们讨论了 Berry 相和手征异常在核物理学和相关领域中的有限数量的代表性应用，以说明本文中提出的这两个基本概念的用法。