We construct a broad class of frustration-free quantum vertex models in 3+1 dimensions (3+1⁢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 ℤ2 gauge constraint enriched with a ℤ2 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⁡(1) 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 ℤ2 deconfined and ℤ2 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.

中文翻译：

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三维量子顶点模型中对称性打破和解限的相互作用


我们在 3+1 维度 （3+1D） 中构建了一大类无挫折量子顶点模型，其基态是经典 3D 顶点模型配置的加权叠加。我们的结果适用于菱形、立方和 BCC 晶格，但通常适用于具有偶数配位数的 3D 晶格。相应的经典顶点模型具有 Z2 规范约束，其中丰富了 Z2 全局对称性。我们通过利用精确的波函数对偶性和有效场论来研究这些对称性之间的相互作用。我们找到了一个精确的无间隙点，它通过对偶性与 U（1） 自旋液体的 Rokhsar-Kivelson （RK） 点有关。此时，对称打破和解限阶参数都表现出长程阶。无间隙点也是第二个对偶性的自对偶点，它将 Z2 去受限和 Z2 对称性断开相相互映射。对于 BCC 晶格顶点模型，我们发现无间隙点靠近对称性打破和解限共存的不寻常中间阶段。