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Aims: We aim to characterise the small-scale magnetic fields of a sample of 16 Sun-like stars and investigate the capabilities of the newly upgraded near-infrared (NIR) instrument CRIRES+ at the Very Large Telescope in the context of small-scale magnetic field studies. Our targets also had their magnetic fields studied with optical spectra, which allowed us to compare magnetic field properties at different spatial scales on the stellar surface and to contrast small-scale magnetic field measurements at different wavelengths.

Methods: We analysed the Zeeman broadening signature for six magnetically sensitive and insensitive Fe I lines in the H-band to measure small-scale magnetic fields on the stellar surfaces of our sample. We used polarised radiative transfer modelling and non-local thermodynamic equilibrium departure coefficients in combination with Markov chain Monte Carlo sampling to determine magnetic field characteristics and non-magnetic stellar parameters. We used two different approaches to describe the small-scale magnetic fields. The first is a two-component model with a single magnetic region and a free magnetic field strength. The second model contains multiple magnetic components with fixed magnetic field strengths.

Results: We found average magnetic field strengths ranging from & SIM;0.4 kG down to < 0.1 kG. The results align closely with other results from high-resolution NIR spectrographs, such as SPIRou. It appears that the typical magnetic field strength in the magnetic region is slightly stronger than 1.3 kG, and for most stars in our sample, this strength is between 1 and 2 kG. We also found that the small-scale fields correlate with the large-scale fields and that the small-scale fields are at least ten times stronger than the large-scale fields inferred with Zeeman Doppler imaging. The two- and multi-component models produce systematically different results, as the strong fields from the multi-component model increase the obtained mean magnetic field strength. When comparing our results with the optical measurements of small-scale fields, we found a systematic offset two to three times stronger than fields in the optical results. This discrepancy cannot be explained by uncertainties in stellar parameters. Care should therefore be taken when comparing results obtained at different wavelengths until a clear cause can be established.