用一维和二维规范场和物质场构造无阻挫哈密顿量的方法
A Recipe for Constructing Frustration-Free Hamiltonians with Gauge and Matter Fields in One and Two Dimensions
摘要：
态和构造，如Kuperberg算法，给出物理系统的配分函数，如格点规范理论，通过将局部张量或权重关联到闭合三角形流形的不同部分，在各个维度上。在这里，我们通过包括物质场来扩展这种构造，以在二维和三维时空中构建配分函数。物质场引入了新的权值，它们对应于一个具有内积的$\mathcal{A}$ -模所描述的Potts自旋组态.在一个有边界的三角形流形上进行这种构造，我们得到了传递矩阵，这些传递矩阵被分解成作用于顶点、链接和格元的局部算子的乘积。顶点和元格算子与Kitaev量子双模型（QDM）中出现的算子相似。链接算子耦合了规范场和物质场，并且它简化为已知模型中常见的相互作用项，例如$\mathbb{Z}_2$规范理论与物质场。传递矩阵导致的哈密顿量是挫折免费的，是完全可解的。根据初始输入、规范群和物质模的选择，我们得到了一种有趣的模型，它具有一种新的基态简并度，这种简并度依赖于在规范作用下物质模中等价类的数目.有些模型有局限的通量激发在散装成为deconfined在表面。这些边缘模式由链路运营商提供的能隙保护。这些属性也出现在“受限的沃克-王”模型，这是3D模型具有有趣的表面状态。除了规范激发之外，物质扇区中也有一些激发，它们是固定的，可以被认为是像伊辛模型中那样的缺陷。本文只考虑玻色物质场。
英文摘要：
State sum constructions, such as Kuperberg's algorithm, give partition functions of physical systems, like lattice gauge theories, in various dimensions by associating local tensors or weights, to different parts of a closed triangulated manifold. Here we extend this construction by including matter fields to build partition functions in both two and three space-time dimensions. The matter fields introduces new weights to the vertices and they correspond to Potts spin configurations described by an $\mathcal{A}$-module with an inner product. Performing this construction on a triangulated manifold with a boundary we obtain the transfer matrices which are decomposed into a product of local operators acting on vertices, links and plaquettes. The vertex and plaquette operators are similar to the ones appearing in the quantum double models (QDM) of Kitaev. The link operator couples the gauge and the matter fields, and it reduces to the usual interaction terms in known models such as $\mathbb{Z}_2$ gauge theory with matter fields. The transfer matrices lead to Hamiltonians that are frustration-free and are exactly solvable. According to the choice of the initial input, that of the gauge group and a matter module, we obtain interesting models which have a new kind of ground state degeneracy that depends on the number of equivalence classes in the matter module under gauge action. Some of the models have confined flux excitations in the bulk which become deconfined at the surface. These edge modes are protected by an energy gap provided by the link operator. These properties also appear in "confined Walker-Wang" models which are 3D models having interesting surface states. Apart from the gauge excitations there are also excitations in the matter sector which are immobile and can be thought of as defects like in the Ising model. We only consider bosonic matter fields in this paper.
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