It is predicted in many theories beyond the standard model that the new interaction relevant to spin is existent. The exchange of an axial vector particle will result in attractive dipole-dipole interaction which can be viewed as an effective magnetic potential that looks quite different from those expected from electromagnetism. In this work, we demonstrate that, instead of the laboratory spin source, stringent constraints can be set on these attractive spin-spin interactions from polarized nuclear matters within neutron stars which have extremely strong magnetic fields (up to 10¹⁵ G in some cases). By considering such an exotic interaction within the framework of relativistic mean field model, we find that the stability of infinite nuclear matter can be influenced significantly when the ratio of coupling strength to boson mass become larger than g_\rm A^2/m_\rm Z'^2 \sim \cal O(10 \, \rm GeV^-2). Furthermore, based on the curvature matrix approach, when g_\rm A^2/m_\rm Z'^2 > 130 \, \rm GeV^-2, phase transition inside low-density nuclear matter will no longer take place before the pressure of nuclear matter becomes zero, which forbids core-crust transition at the inner edge separating the liquid core from the solid crust of neutron stars. Thus bare neutron stars without any crusts are predicted. However, observations of pulsar glitches, i.e., the occasional disruptions of the extremely regular pulsations from magnetized, rotating neutron stars, imply the existence of crusts inside these dense objects. This in turn constrains the strength of the exotic interaction. In fact, in the case of dipole-dipole force on a length scale between µm to cm, the highest value of these constraints can be 8 orders of magnitude higher than those from existing laboratory results.