Topology of generalized spinors and chiral anomaly

Abstract

Weyl fermions with nonlinear dispersion have appeared in real world systems, such as in the Weyl semi-metals and topological insulators. We consider the most general form of Dirac operators, and study its topological properties embedded in the chiral anomaly, in the index theorem, and in the odddimensional partition function, by employing the heat kernel. We find that all of these topological quantities are enhanced by a winding number defined by the Dirac operator in the momentum space, regardless of the spacetime dimensions. The chiral anomaly in d = 3 + 1, in particular, is also confirmed via the conventional Feynman diagram. These interconnected results allow us to clarify the relationship between the chiral anomaly and the Chern number of the Berry connection, under dispute in some recent literatures, and also lead to a compact proof of the Nielsen-Ninomiya theorem.

QCD thermalization: Ab initio approaches and interdisciplinary connections

“…Recent work in this direction includes a Wigner function approach [388][389][390][391][392], chiral effective field theory [393], and a worldline formalism [394]. An important question to resolve in this context is the relation of the dynamics of Berry's phase to that of the chiral anomaly [395][396][397]. A common goal of these approaches is a consistent framework to describe anomalous transport in QCD that can be matched to an anomalous relativistic hydrodynamic description at late times [398].…”

Chiral anomaly in non-relativistic systems: Berry curvature and chiral kinetic theory

“…It would be also interesting to make a field-theoretical analysis for Hamiltonians (1) and ( 2) to reveal the connection of chiral anomaly with ultraviolet divergence in field theory. Such an analysis has been performed for Hamiltonian (1) without K 0 term [55] and will be presented for the case with K 0 in a future work.…”

“…It shows that the term K 0 in Hamiltonian (1) does not change the form of the anomaly relation and contributes to the chiral anomaly only through the distribution function f . We emphasize that the relation (52) can also be understood through the index theorem [55].…”

Chiral Anomaly in Non-Relativistic Systems: Berry Curvature and Chiral Kinetic Theory

“…We emphasize that the relation (48) can also be understood through the index theorem. [55] To end this section, we give the expressions for CME of fermions of type (1) for an equilibrium distribution function which depends only on the energy 𝜀 𝑝 , 𝑓 = 𝑓 (𝜀 𝑝 ),…”

“…It would also be interesting to make a field-theoretical analysis for Hamiltonians (1) and (2) to reveal the connection of chiral anomaly with ultraviolet divergence in field theory. Such an analysis has been performed for Hamiltonian (1) without 𝐾 0 term [55] and will be presented for the case with 𝐾 0 in a future work.…”