We construct a broad class of frustration-free quantum vertex models in $3+1$ dimensions ($3+1\mathrm{D}$) whose ground states are weighted superpositions of classical 3D vertex model configurations. Our results are illustrated for diamond, cubic, and BCC lattices, but hold in general for 3D lattices with even coordination number. The corresponding classical vertex models have a ${\mathbb{Z}}_2$ gauge constraint enriched with a ${\mathbb{Z}}_2$ global symmetry. We study the interplay between these symmetries by exploiting exact wave function dualities and effective field theories. We find an exact gapless point which by duality is related to the Rokhsar-Kivelson (RK) point of $U\left(1\right)$ spin liquids. At this point, both the symmetry breaking and deconfinement order parameters exhibit long range order. The gapless point is additionally a self-dual point of a second duality that maps the ${\mathbb{Z}}_2$ deconfined and ${\mathbb{Z}}_2$ symmetry-broken phases to one another. For the BCC lattice vertex model, we find that gapless point is proximate to an unusual intermediate phase where symmetry breaking and deconfinement coexist.

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• Quantum criticality
• Quantum spin liquid
• Topological order
• Topological phases of matter
• Integrable systems