Berry’s phase and chiral anomalies

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.