Spontaneous wandering of the magnetic axis in pulsars: 3D magneto-thermal simulations and the imprint on braking indices

The braking index measurements of hundreds of pulsars, despite the observational caveats, show a clear trend: n<3 at early ages, n>3 at middle ages, and increasingly large, fluctuating values at larger characteristic ages. Crustal magnetic fields in isolated neutron stars evolve through Ohmic dissipation and non-linear Hall redistribution across scales, with additional early-time contributions from the chiral instability and from post-burial re-emergence. Initial conditions are usually assumed dipole-dominated, whereas more complex and realistic fields, with energy spread more evenly over a broad range of scales, have not been explored in the context of the braking index. We assess how the internal dynamics of initially non-trivial crustal fields change the dipolar field strength and obliquity, and compare their combined contribution to the braking index n, together with that of the expected alignment, with the observed values. We performed long-term 3D magneto-thermal simulations, for initial configurations ranging from the standard dipole-dominated case to tangled, small-scale-dominated fields. Small-scale-dominated configurations with relatively weak dipolar components tend to show rich dynamics: the dipolar mode is continuously fed by the small scales and may grow or decay depending on the initial spectrum, driving n<3 or n>3, while the obliquity varies on timescales as short as $ ̧al O $(kyr). These spontaneous changes often dominate the alignment torque, and can account for the large, sign-alternating braking indices observed at characteristic ages τ_c yr, unlike the dipole-dominated cases. Small-scale initial conditions can reproduce the large values of n-3, of either sign, observed at late ages. However, short-term magnetospheric fluctuations or superfluid-driven variations of the torque may still be needed to explain the available measurements and the structured timing residuals quantitatively. Future work comparing the observed timing properties in detail could fine-tune the most promising initial configurations.