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The square Bi layers in $A$MnBi$_2$ ($A =$ Sr, Ca) host Dirac fermions whichcoexist with antiferromagnetic order on the Mn sublattice below $T_\mathrm{N} =290\,$K (Sr) and $270\,$K (Ca). We have measured the spin-wave dispersion inthese materials by triple-axis neutron spectroscopy. The spectra showpronounced spin gaps of 10.2(2)$\,$meV (Sr) and 8.3(8)$\,$meV (Ca) and extendto a maximum energy transfer of 61 - 63$\,$meV. The observed spectra can beaccurately reproduced by linear spin-wave theory from an Heisenberg effectivespin Hamiltonian. Detailed global fits of the full magnon dispersion are usedto determine the in-plane and inter-layer exchange parameters as well as on themagnetocrystalline anisotropy constant. To within experimental error we find noevidence that the magnetic dynamics are influenced by the Dirac fermions.

Spindynamics in the antiferromagnetic phases of the Dirac metals $A$MnBi$_2$ ($A=$ Sr, Ca)