  I investigate the motion of a single hole in 2D spin lattices with square and
triangular geometries. While the spins have nearest neighbor Ising spin
couplings $J$, the hole is allowed to move only in 1D along a single line in
the 2D lattice with nearest neighbor hopping amplitude $t$. The non-equilibrium
hole dynamics is initialized by suddenly removing a single spin from the
thermal Ising spin lattice. I find that for any nonzero spin coupling and
temperature, the hole is localized. This is an extension of the thermally
induced localization phenomenon [arXiv:2310.11193] to the case, where there is
a phase transition to a long-range ordered ferromagnetic phase. The dynamics
depends only on the ratio of the temperature to the spin coupling, $k_BT /
|J|$, and on the ratio of the spin coupling to the hopping $J/t$. I
characterize these dependencies in great detail. In particular, I find
universal behavior at high temperatures, common features for the square and
triangular lattices across the Curie temperatures for ferromagnetic
interactions, and highly distinct behaviors for the two geometries in the
presence of antiferromagnetic interactions due geometric frustration in the
triangular lattice.
