Disorder from order among anisotropic next-nearest-neighbor Ising spin chains in SrHo$_2$O$_4$

We describe why Ising spin chains with competing interactions in $rm SrHo_2O_4$ segregate into ordered and disordered ensembles at low temperatures ($T$). Using elastic neutron scattering, magnetization, and specific heat measurements, the two distinct spin chains are inferred to have Neel ($uparrowdownarrowuparrowdownarrow$) and double-Neel ($uparrowuparrowdownarrowdownarrow$) ground states respectively. Below $T_mathrm{N}=0.68(2)$~K, the Neel chains develop three dimensional (3D) long range order (LRO), which arrests further thermal equilibration of the double-Neel chains so they remain in a disordered incommensurate state for $T$ below $T_mathrm{S}= 0.52(2)$~K. $rm SrHo_2O_4$ distills an important feature of incommensurate low dimensional magnetism: kinetically trapped topological defects in a quasi$-d-$dimensional spin system can preclude order in $d+1$ dimensions.