Quantum spin liquids host novel emergent excitations, such as monopoles of an emergent gauge field. Here, we study the hierarchy of monopole operators that emerges at quantum critical points (QCPs) between a two-dimensional Dirac spin liquid and various ordered phases. This is described by a confinement transition of quantum electrodynamics in two spatial dimensions (QED₃ Gross–Neveu theories). Focusing on a spin ordering transition, we get the scaling dimension of monopoles at leading order in a large-$N$ expansion, where $2N$ is the number of Dirac fermions, as a function of the monopole’s total magnetic spin. Monopoles with a maximal spin have the smallest scaling dimension while monopoles with a vanishing magnetic spin have the largest one, the same as in pure QED₃. The organization of monopoles in multiplets of the QCP’s symmetry group SU(2) $\times$ SU(N) is shown for general N.

量子自旋液体具有新颖的出射激发，例如出射规矩场的单极子。在这里，我们研究在二维Dirac自旋液体和各种有序相之间的量子临界点（QCP）处出现的单极算子的层次。这是通过量子电动力学在两个空间维度上的局限转变来描述的（QED ₃ Gross-Neveu理论）。着眼于自旋有序过渡，我们获得了在大范围内处于领先地位的单极子的缩放尺度，$ñ$ 扩展，在哪里 $\mathrm{2个}ñ$是狄拉克费米子数，它是单极子总磁自旋的函数。自旋最大的单极具有最小的缩放尺度，而自旋消失的单极具有最大的缩放尺度，与纯QED ₃相同。QCP对称群SU（2）的多重态中的单极组织$\times$ SU（N）显示为一般N。