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

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.