The disorder operator, as an easily measured nonlocal observable, displays great potential in detecting intrinsic information of field theories. It has been systematically studied in one- and two-dimensional (1D and 2D) quantum systems, while the knowledge of 3D is still limited. The disorder operator associated with U(1) global symmetry exhibits rich geometric dependence on the shape of the spatial region at a quantum critical point, meanwhile, $(3+1)\mathrm{D}$ is the upper critical dimension for $O\left(N\right)$ criticality, both of which pose a challenge for exploring the disorder operator in high dimensions. In this Letter, we investigate the scaling behaviors of disorder operators in $(3+1)\mathrm{D}$ $O\left(3\right)$ models through a large-scale quantum Monte Carlo simulation combined with a theoretical analysis. Although the upper critical dimension introduces logarithmic corrections in correlations, we analytically prove and numerically demonstrate that the corrections do not modify the universal trihedral-corner contribution of the disorder operator. The universal contributions, such as the current central charge, have been revealed in our calculation, which establishes a concrete link between lattice simulations and continuum field theory. This work opens promising directions for the experimental and numerical exploration of universal properties at quantum critical points in $(3+1)\mathrm{D}$ models.

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• Magnetism
• Quantum phase transitions
• Conformal field theory
• Heisenberg model
• Quantum Monte Carlo