This review describes how topological order associated with the presence of emergent gauge fields can reconstruct Fermi surfaces of metals, even in the absence of translational symmetry breaking. We begin with an introduction to topological order using Wegner's quantum [Formula: see text] gauge theory on the square lattice: the topological state is characterized by the expulsion of defects, carrying [Formula: see text] magnetic flux. The interplay between topological order and the breaking of global symmetry is described by the non-zero temperature statistical mechanics of classical XY models in dimension D = 3; such models also describe the zero temperature quantum phases of bosons with short-range interactions on the square lattice at integer filling. The topological state is again characterized by the expulsion of certain defects, in a state with fluctuating symmetry-breaking order, along with the presence of emergent gauge fields. The phase diagrams of the [Formula: see text] gauge theory and the XY models are obtained by embedding them in U(1) gauge theories, and by studying their Higgs and confining phases. These ideas are then applied to the single-band Hubbard model on the square lattice. A SU(2) gauge theory describes the fluctuations of spin-density-wave order, and its phase diagram is presented by analogy to the XY models. We obtain a class of zero temperature metallic states with fluctuating spin-density wave order, topological order associated with defect expulsion, deconfined emergent gauge fields, reconstructed Fermi surfaces (with 'chargon' or electron-like quasiparticles), but no broken symmetry. We conclude with the application of such metallic states to the pseudogap phase of the cuprates, and note the recent comparison with numerical studies of the Hubbard model and photoemission observations of the electron-doped cuprates. In a detour, we also discuss the influence of Berry phases, and how they can lead to deconfined quantum critical points: this applies to bosons on the square lattice at half-integer filling, and to quantum dimer models.

中文翻译：

---

拓扑序、涌现规范场和费米面重建

这篇综述描述了与涌现规范场的存在相关的拓扑顺序如何重建金属的费米面，即使在没有平移对称性破缺的情况下。我们首先使用 Wegner 的量子 [公式：见正文] 规范理论对方形晶格的拓扑序进行介绍：拓扑状态的特征是排除缺陷，携带 [公式：见正文] 磁通量。拓扑顺序与全局对称性破坏之间的相互作用由维度 D = 3 的经典 XY 模型的非零温度统计力学描述；此类模型还描述了玻色子在整数填充时在方形晶格上具有短程相互作用的零温度量子相。拓扑状态再次以排除某些缺陷为特征，处于对称破缺阶次波动的状态，以及涌现规范场的存在。[公式：见正文]规范理论和 XY 模型的相图是通过将它们嵌入 U(1) 规范理论中，并通过研究它们的希格斯和限制相而获得的。然后将这些想法应用于方形点阵上的单波段哈伯德模型。SU(2)规范理论描述了自旋-密度-波阶的涨落，其相图通过类比XY模型来表示。我们获得了一类具有波动自旋密度波序、与缺陷驱逐相关的拓扑序、去约束的涌现规范场、重建的费米表面（带有“电荷”或类电子准粒子）的零温度金属态，但没有破坏对称性。我们最后将这种金属态应用于铜酸盐的赝隙相，并注意最近与哈伯德模型的数值研究和电子掺杂铜酸盐的光电发射观察的比较。绕道而行，我们还讨论了 Berry 相的影响，以及它们如何导致去限制的量子临界点：这适用于半整数填充的方形晶格上的玻色子，以及量子二聚体模型。