Haldane's Instanton in 2D Heisenberg model revisited: along the Avenue of Topology

TL;DR

This paper revisits the role of Haldane's instanton in the 2D Heisenberg model using topological methods, providing new insights into monopole suppression and the Berry phase's effects on quantum criticality.

Contribution

It introduces a topological perspective to analyze Haldane's instanton, linking monopole events to topological charges and clarifying conditions for their suppression.

Findings

Monopole events are space-time singularities of the Néel field.

Suppression of monopoles requires the $oldsymbol{}$-field to have no zero points.

Berry phase induces quadrupolarity of monopole events in the model.

Abstract

Deconfined quantum phase transition from N\'eel phase to Valence bond crystal state in 2D Heisenberg model is under debate nowadays. One crucial issue is the suppression of Haldane's instanton on quantum critical point which drives the spinon deconfined. In this paper, by making use of the $\phi$-mapping topological current theory, we reexamine the Haldane's instanton in an alternative way along the direction of topology. We find that the the monopole events are space-time singularities of N\'eel field $\vec{n}$, the corresponding topological charges are the wrapping number of $\vec{n}$ around the singularities which can be expressed in terms of the Hopf indices and Brouwer degrees of $\phi$-mapping. The suppression of the monopole events can only be guaranteed when the $\phi$-field possesses no zero points. Moreover, the quadrapolarity of monopole events in the Heisenberg model due to the Berry phase is also reproduced in this topological argument.