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We construct a broad class of frustration-free quantum vertex models in 3+1Dwhose ground states are weighted superpositions of classical 3D vertex modelconfigurations. For the simplest members of this class, the correspondingclassical vertex models have a $\mathbb{Z}_2$ gauge constraint enriched with a$\mathbb{Z}_2$ global symmetry. We deduce a phase diagram for the quantumvertex models using exact dualities and mappings to classical spin models andheight models. We find $\mathbb{Z}_2$ deconfined and $\mathbb{Z}_2$symmetry-broken phases which are mapped to one another by duality. Thetransition between these phases is governed by a self-dual gapless point whichis related by a different duality to the Rokhsar-Kivelson (RK) point of quantumspin ice. Our results are illustrated for diamond and cubic lattices, but holdfor general 3D lattices with even coordination number.

Exact wavefunction dualities and phase diagrams of 3D quantum vertex models