We present a solvable one-dimensional spin-1/2 model on the diamond chain featuring three-spin interactions, which displays both, mobile excitations driving a second-order phase transition between an ordered and a Z₂ ℤ 2 -symmetry broken phase, as well as non-trivial fully immobile excitations. The model is motivated by the physics of fracton excitations, which only possess mobility in a reduced dimension compared to the full model. We provide an exact mapping of this model to an arbitrary number of independent transverse-field Ising chain segments with open boundary conditions. The number and lengths of these segments correspond directly to the number of immobile excitations and their respective distances from one another. Furthermore, we demonstrate that multiple immobile excitations exhibit Casimir-like forces between them, resulting in a non-trivial spectrum.
Bayer et al. (2026) studied this question.