Multiwavelength observations of PSR J2021+4026 across a mode change reveal a phase shift in its X-ray emission

Context. We have investigated the multiwavelength emission of PSR J2021+4026, the only isolated gamma-ray pulsar known to be variable, which in October 2011 underwent a simultaneous change in gamma-ray flux and spin-down rate, followed by a second mode change in February 2018. Multiwavelength monitoring is crucial to understand the physics behind these events and how they may have affected the structure of the magnetosphere. Aims.The monitoring of pulse profile alignment is a powerful diagnostic tool for constraining magnetospheric reconfiguration. We aim to investigate timing or flux changes related to the variability of PSR J2021+4026 via multiwavelength observations, including gamma-ray observations from Fermi-LAT, X-ray observations from XMM-Newton, and a deep optical observation with the Gran Telescopio Canarias.Methods. We performed a detailed comparison of the timing features of the pulsar in gamma and X-rays and searched for any change in phase lag between the phaseogram peaks in these two energy bands. Although previous observations did not detect a counterpart in visible light, we also searched for optical emission that might have increased due to the mode change, making this pulsar detectable in the optical. Results.We have found a change in the gamma-to X-ray pulse profile alignment by 0.21$\pm$0.02 in phase, which indicates that the first mode change affected different regions of the pulsar magnetosphere. No optical counterpart was detected down to g'=26.1 and r'=25.3. Conclusions.We suggest that the observed phase shift could be related to a reconfiguration of the connection between the quadrupole magnetic field near the stellar surface and the dipole field that dominates at larger distances. This is consistent with the picture of X-ray emission coming from the heated polar cap and with the simultaneous flux and frequency derivative change observed during the mode changes.