The Ising chain in a magnetic field is solved exactly. Adding lattice deformability — letting the spin interactions depend on how much the lattice stretches — makes the model more realistic but potentially intractable. The magnetoelastic coupling introduces feedback: spins affect the lattice; the lattice affects the spins.
The model remains exactly solvable (arXiv:2603.06125). Transfer-matrix and Jordan-Wigner fermionization techniques yield the variational Gibbs free energy, optimized over distortion parameters. Under longitudinal fields, the chain exhibits discontinuous thermal phase transitions terminating at a critical point, with hysteresis. Under transverse fields, only a continuous quantum phase transition occurs at zero temperature.
The structural observation: the phase transitions announce themselves through elastic properties — sound attenuation and elastic softening — before they manifest magnetically. The lattice, which is coupled to the spins, softens near the transition because the effective spring constant depends on spin correlations that diverge. The elastic channel carries the signal of the approaching transition while the magnetic channel is still far from singular. The lattice is not just a passive substrate for the spins; it is a diagnostic instrument that detects the transition through a different observable than the one undergoing it.