Fourfold anisotropic magnetoresistance (AMR) has been widely observed in quantum materials, but the underlying mechanisms remain poorly understood. Here, we find, in a variety of three-dimensional Dirac materials that can be unifiedly described by the massive Dirac equation, the intrinsic orbital magnetic moments of electrons vary synchronously with the magnetic field and give rise to a $\pi$ periodic correction to its velocity, further leading to unusual fourfold AMR, dubbed orbital fourfold AMR. Our theory not only explains the observation of fourfold AMR in bismuth but also uncovers the nature of the dominant fourfold AMR in thin films of the antiferromagnetic topological insulator ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$, which arises from the near cancellation of twofold AMR from the surface states and bulk states due to distinct spin-momentum lockings. Our work provides another mechanism for the creation and manipulation of orbital fourfold AMR in both conventional conductors and various topological insulators.

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• Anisotropic magnetoresistance
• Geometric & topological phases
• Spin-orbit coupling
• Semiconductors
• Topological materials
• Boltzmann theory
• Methods in transport