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 $D$.

Figure 1

Lattice used to derive continuum Dirac theory.

Figure 2

Sequence of $\frac{\pi }{2}$ rotations around a dual site which simultaneously rotates the direct sites and the VBS order parameter.