We investigate a deformable spin-1/2 Ising chain subjected to either a longitudinal or a transverse magnetic field, which incorporates a magnetoelastic coupling linearly dependent on a lattice distortion parameter. Within the harmonic and static adiabatic approximations, the variational Gibbs free energy is evaluated exactly using transfer-matrix and Jordan-Wigner fermionization techniques and then minimized self-consistently with respect to the lattice distortion parameter. This approach enables a unified description of magnetic and elastic properties including the magnetization, magnetic susceptibility, lattice distortion, inverse compressibility, and relative change in the sound velocity. In a longitudinal magnetic field, the deformable Ising chain displays a line of discontinuous thermal phase transitions terminating at a critical point. The discontinuous transitions are accompanied by metastable states, which give rise to a hysteresis loop at low temperatures. In contrast, the deformable Ising chain in a transverse field undergoes exclusively a continuous quantum phase transition at zero temperature with no indication of thermal phase transitions. The magnetic susceptibility and inverse compressibility exhibit cusp- and dip-like anomalies at discontinuous phase transitions, while a diverging susceptibility and vanishing inverse compressibility characterize the continuous phase transitions. An elastic softening of the deformable chain near thermal and quantum phase transitions manifests itself also through a significant sound attenuation.

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• Phase transitions
• Quantum phase transitions
• Lattice models in statistical physics
• Exact solutions for many-body systems
• Ising model
• Transfer matrix calculations