The emergence of Hopf term and the marginality of Dirac model

TL;DR

This paper explores how the marginality of the Dirac model relates to the emergence of the Hopf term, highlighting the role of non-relativistic terms and boundary modes in this process.

Contribution

It reveals the necessity of infinitesimal non-relativistic terms for the Hopf term emergence and links boundary mode chirality to coupling sign.

Findings

Hopf term emergence requires marginality improvement of Dirac modes.

Chiral boundary modes always accompany the Hopf term.

Sign of non-relativistic coupling determines boundary mode chirality.

Abstract

The subtle relation of the marginality of Dirac model and the emergence of the Hopf term through a Yukawa-type interaction is revealed in this work. We show that the improvement of the marginality of Dirac mode through an infinitesimal non-relativistic term, which is irrelevant for renormalization group, is necessary to emerge the Hopf term. It is found that the appearance of the Hopf term is always accompanied with chiral boundary modes, and the sign of the coupling constant of the infinitesimal non-relativistic term decides their chirality. A lattice Dirac model is also constructed to realize the Hopf term.