  For about half a century the radio pulsar population was observed to spin in
the ~0.002-12s range, with different pulsar classes having a spin-period
evolution that differs substantially depending on their magnetic fields or past
accretion history. The recent detection of several slowly rotating pulsars has
re-opened the long-standing question of the exact physics, and observational
biases, driving the upper bound of the period range of the pulsar population.
In this work, we perform a parameter study of the spin-period evolution of
pulsars interacting with supernova fallback matter and specifically look at the
fallback accretion disk scenario. Depending on the initial conditions at
formation, this evolution can differ substantially from the typical dipolar
spin-down, resulting in pulsars that show spin periods longer than their coeval
peers. By using general assumptions for the pulsar spin period and magnetic
field at birth, initial fallback accretion rates and including magnetic field
decay, we find that very long spin periods (>100s) can be reached in the
presence of strong, magnetar-like magnetic fields (>10^{14}G) and moderate
initial fallback accretion rates (~10^{22-27} g s^{-1}). In addition, we study
the cases of two recently discovered periodic radio sources, the pulsar PSR
J0901-4046 (P = 75.9 s) and the radio transient GLEAM-X\,J162759.5-523504.3 (P
= 1091 s), in light of our model. We conclude that the supernova fallback
scenario could represent a viable channel to produce a population of
long-period isolated pulsars that only recent observation campaigns are
starting to unveil.
