Determination of the Range of Magnetic Interactions from the Relations between Magnon Eigenvalues at High-Symmetry k Points

Magnetic exchange interactions (MEIs) define networks of coupled magnetic moments and lead to a surprisingly rich variety of their magnetic properties. Typically MEIs can be estimated by fitting experimental results. Unfortunately, how many MEIs need to be included in the fitting process for a material is unclear a priori, which limits the results obtained by these conventional methods. Based on linear spin-wave theory but without performing matrix diagonalization, we show that for a general quadratic spin Hamiltonian, there is a simple relation between the Fourier transform of MEIs and the sum of square of magnon energies (SSME). We further show that according to the real-space distance range within which MEIs are considered relevant, one can obtain the corresponding relationships between SSME in momentum space. By directly utilizing these characteristics and the experimental magnon energies at only a few high-symmetry k points in the Brillouin zone, one can obtain strong constraints about the range of exchange path beyond which MEIs can be safely neglected. Our methodology is also generally applicable for other Hamiltonian with quadratic Fermi or Boson operators.