Many-body spin Berry phases emerging from the pi-flux state: competition between antiferromagnetism and the valence-bond-solid state.

We uncover new topology-related features of the pi-flux saddle-point solution of the D=2+1 Heisenberg antiferromagnet. We note that symmetries of the spinons sustain a built-in competition between antiferromagnetic (AFM) and valence-bond-solid (VBS) orders, the two tendencies central to recent developments on quantum criticality. An effective theory containing an analogue of the Wess-Zumino-Witten term is derived, which generates quantum phases related to AFM monopoles with VBS cores, and reproduces Haldane's hedgehog Berry phases. The theory readily generalizes to pi-flux states for all D.