We study the low-energy effective theory of spin-1/2 quantum ice on the planar pyrochlore lattice from the perspective of its global symmetries. We show that the Fourier transform of the spin operator $S_{\mathbf{Q}}^z$ can be expressed in terms of conserved quantities associated with a one-form U(1) symmetry along two high-symmetry momentum paths, leading to constraints on both the static and dynamical structure factors. We also derive Lieb–Schultz–Mattis type constraints that forbid the existence of a unique gapped ground state in the low-energy effective theory. Our findings also apply to the $(2+1)$-dimensional U(1) quantum link model.

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• Spin ice
• Symmetries
• Lattice gauge theory