Topological states of matter are, generally, quantum liquids of conserved topological defects. We establish this by constructing and analyzing topological field theories which introduce gauge fields to describe the dynamics of singularities in the original field configurations. Homotopy groups are utilized to identify topologically protected singularities, and the conservation of their protected number is captured by a topological action term that unambiguously obtains from the given set of symmetries. Stable phases of these theories include quantum liquids with emergent massless Abelian and non-Abelian gauge fields, as well as topological orders with long-range quantum entanglement, fractional excitations, boundary modes, and unconventional responses to external perturbations. This paper focuses on the derivation of topological field theories and basic phenomenological characterization of topological orders associated with homotopy groups ${\pi }_n\left(S^n\right), n\ge 1$. These homotopies govern monopole and hedgehog topological defects in $d=n+1$ dimensions, and enable the generalization of both weakly interacting and fractional quantum Hall liquids of vortices to $d>2$. Hedgehogs have not been in the spotlight so far, but they are particularly important defects of magnetic moments because they can be stimulated in realistic systems with spin-orbit coupling, such as chiral magnets and $d=3$ topological materials. We predict topological orders in systems with $\mathrm{U}\left(1\right)\times \mathrm{Spin}\left(d\right)$ symmetry in which fractional electric charge attaches to hedgehogs. Monopoles, the analogous defects of charge or generic U(1) currents, may bind to hedgehogs via Zeeman effect, or effectively emerge in purely magnetic systems. The latter can lead to spin liquids with different topological orders than that of the resonant valence bond spin liquid. Charge fractionalization of quarks in atomic nuclei is also seen as possibly arising from the charge-hedgehog attachment.

中文翻译：

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单极子和刺猬的拓扑顺序：从电子和磁性自旋轨道耦合到夸克

物质的拓扑状态通常是守恒拓扑缺陷的量子液体。我们通过构建和分析拓扑场理论来建立这种关系，这些理论引入了规范场来描述原始场配置中奇异点的动力学。利用同伦基团来识别拓扑受保护的奇点，并通过从给定的对称集合中明确获得的拓扑作用项来捕获其受保护数的守恒。这些理论的稳定阶段包括具有新兴的无质量阿贝尔和非阿贝尔规范场的量子液体，以及具有长距离量子纠缠，分数激发，边界模式和对外部扰动的非常规响应的拓扑阶。${\pi }_{ñ}（{\mathrm{小号}}^{ñ}）， ñ\ge \mathrm{1个}$。这些同伦控制了单极和刺猬的拓扑缺陷。$d=ñ+\mathrm{1个}$ 尺寸，并使旋涡的弱相互作用和分数量子霍尔液体都可以推广到 $d>2$。到目前为止，刺猬还没有引起人们的注意，但是它们是磁矩的特别重要的缺陷，因为它们可以在具有自旋轨道耦合的现实系统中被激发，例如手性磁铁和$d=3$拓扑材料。我们预测具有以下特征的系统中的拓扑顺序$\mathrm{ü}（\mathrm{1个}）\times \mathrm{旋转}（d）$分数电荷附着在刺猬上的对称性。单极子是电荷或通用U（1）电流的类似缺陷，可能通过塞曼效应与刺猬结合，或者在纯磁系统中有效地出现。后者可以导致与共振价键自旋液体相比具有不同拓扑顺序的自旋液体。夸克在原子核中的电荷分级也被认为可能是由电荷-刺猬附着引起的。