Pyrochlore photons: The U ( 1 ) spin liquid in a S = 1 2 three-dimensional frustrated magnet

We study the $S=1/2$ Heisenberg antiferromagnet on the pyrochlore lattice in the limit of strong easy-axis exchange anisotropy. We find, using only standard techniques of degenerate perturbation theory, that the model has a $U(1)$ gauge symmetry generated by certain local rotations about the z axis in spin space. Upon addition of an extra local interaction in this and a related model with spins on a three-dimensional network of corner-sharing octahedra, we can write down the exact ground-state wave function with no further approximations. Using the properties of the soluble point we show that these models enter the $U(1)$ spin liquid phase, a fractionalized spin liquid with an emergent $U(1)$ gauge structure. This phase supports gapped ${S}^{z}=1/2$ spinons carrying the $U(1)$ ``electric'' gauge charge, a gapped topological point defect or ``magnetic'' monopole, and a gapless ``photon,'' which in spin language is a gapless, linearly dispersing ${S}^{z}=0$ collective mode. There are power-law spin correlations with a nontrivial angular dependence, as well as $U(1)$ topological order. This state is stable to all zero-temperature perturbations and exists over a finite extent of the phase diagram. Using a convenient lattice version of electric-magnetic duality, we develop the effective description of the $U(1)$ spin liquid and the adjacent soluble point in terms of Gaussian quantum electrodynamics and calculate a few of the universal properties. The resulting picture is confirmed by our numerical analysis of the soluble point wave function. Finally, we briefly discuss the prospects for understanding this physics in a wider range of models and for making contact with experiments.