Altermagnetic (AM) states have compensated collinear magnetic configurations that are invariant under a combination of real-space rotation and time reversal. While these symmetries forbid a direct bilinear coupling of the AM order parameter with a magnetic field, they generally enable piezomagnetism, manifested as a trilinear coupling with the magnetic field and strain. Here, we show that, as a result of this coupling, in an altermagnet subjected to random strain, the magnetic field triggers an effective random field conjugate to the AM order parameter, providing a rare realization of a tunable random-field Ising model. Specifically, we find two competing effects promoted by an external magnetic field: an increasing random-field disorder, which suppresses long-range AM order, and an enhanced coupling to elastic fluctuations, which favors AM order. By solving the corresponding random-field transverse-field Ising model via a mean-field approach, we obtain the temperature-magnetic field phase diagram of an inhomogeneous AM state for different strengths of random-strain disorder, unveiling the emergence of a field-induced reentrant AM phase. We also discuss the fingerprints of this rich behavior on several experimentally-accessible quantities, such as the shear modulus, the elastocaloric effect coefficient, and the AM order parameter. Our results reveal an unusual but experimentally feasible path to tune AM order with uniform magnetic fields.

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• Altermagnetism
• Magnetic order
• Magnetism
• Nematic order
• Strain
• Altermagnets
• Disordered systems