Incommensurate broken helix induced by nonstoichiometry in the axion insulator candidate EuIn$_{2}$As$_{2}$

Zintl phase EuIn$_{2}$As$_{2}$ has garnered growing attention as an axion insulator candidate, triggered by the identification of a commensurate double-${\mathbf Q}$ broken-helix state in previous studies, however, its periodicity and symmetry remain subjects of debate. Here, we perform resonant x-ray scattering experiments on EuIn$_{2}$As$_{2}$, revealing an incommensurate nature of the broken-helix state, where both the wave number and the amplitude of the helical modulation exhibit systematic sample dependence. Furthermore, the application of an in-plane magnetic field brings about a fanlike state that appears to preserve the double-${\mathbf Q}$ nature, which might be attributed to multiple-spin interactions in momentum space. We propose that the itinerant character of EuIn$_{2}$As$_{2}$, most likely induced by Eu deficiency, gives rise to the helical modulation and impedes the realization of a theoretically-predicted axion state with the collinear antiferromagnetic order.