Monopole operators are studied in a large family of quantum critical points between Dirac spin liquids and topological quantum spin liquids (QSLs): chiral and $Z_2$ QSLs. These quantum phase transitions are described by conformal field theories (CFTs): quantum electrodynamics in $2+1$ dimensions with $2N$ flavors of two-component massless Dirac fermions and a four-fermion interaction term. For the transition to a chiral spin liquid, it is the Gross-Neveu interaction (${\mathrm{QED}}_3\text{−}\mathrm{GN}$), while for the transitions to $Z_2$ QSLs, it is a superconducting pairing term with general spin and valley structure (generalized ${\mathrm{QED}}_3\text{−}{Z}_2\mathrm{GN}$). Using the state-operator correspondence, we obtain monopole scaling dimensions to subleading order in $1/N$. For monopoles with a minimal topological charge $q=1/2$, the scaling dimension is $2N\times 0.26510$ at leading order, with the quantum correction being 0.118911(7) for the chiral spin liquid, and 0.102846(9) for the simplest $Z_2$ case (the expression is also given for a general pairing term). Although these two anomalous dimensions are nearly equal, the underlying quantum fluctuations possess distinct origins. The analogous result in ${\mathrm{QED}}_3$ is also obtained, and we find a subleading contribution of $-0.038138(5)$, which is slightly different from the value $-0.0383$ first obtained in the literature. The scaling dimension of a ${\mathrm{QED}}_3\text{−}\mathrm{GN}$ monopole with minimal charge is very close to the scaling dimensions of other operators predicted to be equal by a conjectured duality between ${\mathrm{QED}}_3\text{−}\mathrm{GN}$ with $2N=2$ flavors and the ${\mathrm{C}\mathrm{P}}^1$ model. Additionally, nonminimally charged monopoles with equal charges on both sides of the duality have similar scaling dimensions. By studying the large-$q$ asymptotics of the scaling dimensions in ${\mathrm{QED}}_3$, ${\mathrm{QED}}_3\text{−}\mathrm{GN}$, and ${\mathrm{QED}}_3\text{−}{Z}_2\mathrm{GN}$, we verify that the constant $O(q^0)$ coefficient precisely matches the universal nonperturbative prediction for CFTs with a global U(1) symmetry. Finally, we identify numerous open questions regarding the fate of monopoles and their hierarchies at transitions to spin liquids and ordered phases.

中文翻译：

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在狄拉克和拓扑自旋液体之间转换的单极算子的反常维数

在狄拉克自旋液体和拓扑量子自旋液体 (QSL) 之间的一大类量子临界点中研究了单极子算子：手性和$Z_2$QSL。这些量子相变由共形场论 (CFT) 描述：量子电动力学$2+1$尺寸与$2ñ$双分量无质量狄拉克费米子和四费米子相互作用项的味道。对于向手性自旋液体的转变，它是 Gross-Neveu 相互作用 (${\mathrm{量子点}}_3\text{-}\mathrm{GN}$)，而对于过渡到$Z_2$QSLs，它是一个具有一般自旋和谷结构的超导配对项（广义${\mathrm{量子点}}_3\text{-}{Z}_2\mathrm{GN}$）。使用状态算子对应关系，我们获得了单极子缩放维度到次领先的顺序$1/ñ$. 对于具有最小拓扑电荷的单极子$q=1/2$，缩放维度为$2ñ\times 0.26510$处于领先地位，手性自旋液体的量子校正为 0.118911(7)，最简单的量子校正为 0.102846(9)$Z_2$case （该表达式也用于一般配对项）。尽管这两个异常维度几乎相等，但潜在的量子涨落具有不同的起源。类似的结果${\mathrm{量子点}}_3$也获得了，我们发现了一个次要贡献$-0.038138(5)$，与值略有不同$-0.0383$最早在文献中获得。a的缩放维度${\mathrm{量子点}}_3\text{-}\mathrm{GN}$具有最小电荷的单极子非常接近其他算子的缩放维度，这些算子通过推测的对偶性预测为相等${\mathrm{量子点}}_3\text{-}\mathrm{GN}$和$2ñ=2$味道和${\mathrm{C}\mathrm{磷}}^1$模型。此外，对偶两侧具有相等电荷的非最小带电单极子具有相似的缩放尺寸。通过研究大$q$缩放维度的渐近线${\mathrm{量子点}}_3$,${\mathrm{量子点}}_3\text{-}\mathrm{GN}$， 和${\mathrm{量子点}}_3\text{-}{Z}_2\mathrm{GN}$，我们验证常数$○(q^0)$系数与具有全局 U(1) 对称性的 CFT 的通用非微扰预测精确匹配。最后，我们确定了许多关于单极子的命运及其在自旋液体和有序相转变中的层次结构的开放性问题。