Rotational modulation and long-term evolution of the small-scale magnetic fields of M dwarfs observed with SPIRou

M dwarfs are known to host magnetic fields, impacting exoplanet studies and playing a key role in stellar and planetary formation and evolution. Observations revealed the long-term evolution of the large-scale magnetic field reconstructed with Zeeman-Doppler imaging, and a diversity of their topologies. These large-scale magnetic fields only account for a small amount of the unsigned magnetic flux that can be probed by directly modeling the Zeeman broadening of spectral lines in unpolarized spectra. We aim at investigating the long-term behavior of the average small-scale magnetic field of M dwarfs with time, and assess our ability to detect rotational modulation from time series of field measurements derived from unpolarized spectra. We perform fits of synthetic spectra computed with ZeeTurbo to near-infrared high-resolution spectra recorded with SPIRou between 2019 and 2024 in the context of the SLS and SPICE large programs. The analysis is performed on the spectra of 2 partially convective (AD Leo, DS Leo) and 3 fully convective (PM J18482+0741, CN Leo, Barnard star) M dwarfs, along with EV Lac whose mass is close to the fully-convective limit. Our analysis provides measurements of the average small-scale magnetic field, which are compared to longitudinal magnetic field and temperature variation measurements (d$Temp$) obtained from the same data. We were able to detect the rotation period in the small-scale magnetic field series for 4 of the 6 stars in our sample. We find that the average magnetic field can vary by up to 0.3 kG throughout the year (e.g., CN Leo), or of up to 1 kG across rotation phases. The rotation periods retrieved from longitudinal and small-scale magnetic fields are found in agreement within error bars. d$Temp$ measurements are found to anti-correlate with small-scale magnetic field measurements for three stars (EV Lac, DS Leo and Barnard's star).