$$\widehat{P}\widehat{{T}}$$ symmetry controlled magnetic order switching

Precise electric control of magnetic order and anomalous Hall conductivity (AHC) is pivotal for spintronics. While electric-field control of magnetic order and AHC has been explored in magneto-electric materials, achieving precise and energy-efficient magnetic order switching between two \(\hat{P}\hat{{T}}\) symmetry-connected magnetic states remains challenging. Here, we propose the utilization of the combined \(\widehat{P}\widehat{{T}}\) symmetry that establishes a direct connection between electric polarization and magnetic orders, to electrically manipulate magnetic order and the AHC. Using 3MnB₂T₄·2B₂T₃ (B = Sb/Bi, T = Se/Te) as an example, we demonstrate that the \(\widehat{P}\widehat{{T}}\) connected up-up-down (UUD) and up-down-down (UDD) states exhibit switchable magnetic configurations via electric polarization. The energy difference between the UUD and UDD states is linearly modulated by electric polarizations, enabling full control of the magnetic states via electric field, spontaneous polarization, or even weak sliding ferroelectricity. The findings demonstrate that \(\widehat{P}\widehat{{T}}\) symmetry can be well utilized to design electric polarization-controlled magnetic orders and will find important applications in spintronics.