Topology of generalized spinors and chiral anomaly

Piljin Yi,Ho-Ung Yee

doi:10.1103/physrevd.101.045007

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 odd-dimensional 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.

Highlights

In quantum field theories with fermions, we often encounter topological properties, with the chiral anomaly being perhaps the best-known such effect
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
Can we prove the existence of such an infraredultraviolet connection in momentum space? If yes, what topology of the fermion theory in large momenta contains the information of the infrared chiral anomaly? One of our main results in this work is to provide a rigorous answer to this question

Summary

INTRODUCTION

In quantum field theories with fermions, we often encounter topological properties, with the chiral anomaly being perhaps the best-known such effect. The Berry curvature of chiral spinors is an essential ingredient of the kinetic description of chiral particles in phase space, the chiral kinetic theory [19,20,21,22,23], where semiclassical approximation is justified at large momenta It is responsible for many novel transport phenomena in the real-time dynamics of (pseudo)chiral fermion systems, in both the condensed matter physics of Dirac/Weyl semimetals [19,24,25,26,27,28,29] and the physics of quark-gluon plasma in relativistic heavy-ion collisions [30,31]. IV, we will resort to the usual triangular Feynman diagram for the d 1⁄4 4 chiral anomaly, where the modification of the current operators, on top of the higher inverse power of the propagator, plays a crucial role

GENERALIZED SPINORS AND DIRAC OPERATORS

Index density and chiral anomaly

Γγμγνγμ0 γν0 4

Infrared interpretation

NONRELATIVISTIC ISOSPINOR

The Berry monopoles in the momentum space

Infrared view and real material

DIAGRAMMATIC COMPUTATION OF THE ANOMALY

GENERALIZED SPINORS IN ODD DIMENSIONS

Eta invariant for generalized spinors

Findings

Boundary fermions and the APS-like index theorem