Hydrostatic and chemical pressure driven crossover from the commensurate to the incommensurate state of the Weyl semimetal ${\mathrm{Mn}}_{3+x}{\mathrm{Sn}}_{1-x}$

The observation of large intrinsic anomalous Hall conductivity (AHC) in the noncollinear antiferromagnetic phase of the Weyl semimetal ${\mathrm{Mn}}_3\mathrm{Sn}$ generates enormous interest in uncovering the entanglement between the real-space magnetic ordering and the momentum-space band structure. Previous studies show that changes in the magnetic structure induced by the application of hydrostatic and chemical pressure can significantly affect the AHC of the ${\mathrm{Mn}}_{3+x}{\mathrm{Sn}}_{1-x}$ system. Here, we employ the muon spin relaxation/rotation (${\mu }^{+}\mathrm{SR}$) technique to systematically investigate the evolution of different magnetic states in the ${\mathrm{Mn}}_{3+x}{\mathrm{Sn}}_{1-x}$ as a function of hydrostatic and chemical pressure. We find two muon sites experimentally, which is also supported by our ab initio calculations. Our ${\mu }^{+}\mathrm{SR}$ experiments affirm that the $x=0.05$ compound exhibits a commensurate magnetic state throughout the magnetically ordered phase below the Neel temperature $T_N\approx 420$ K in ambient pressure. In contrast, we observe an incommensurate magnetic state below $T_{IC}\sim 175$ K when a hydrostatic pressure of 1.5 GPa is applied. A similar transition from the commensurate to incommensurate state is also found with chemical pressure for $x=0.04$ and $x=0.03$, using ${\mu }^{+}\mathrm{SR}$ and elastic neutron scattering experiments. Using band structure calculations, we have shown the emergence of Fermi nesting in ${\mathrm{Mn}}_3\mathrm{Sn}$ and the subsequent development of incommensurate magnetic ordering under hydrostatic/chemical pressure.