What measured quantity can be associated to the value of the J parameter in the Heisenberg/Ising hamiltonians?

Here I'll assume that the material is already believed to be roughly described by a model of the Heisenberg or Ising form. In that case, you just want a quantity that is easy to measure in your experiment and easy to extract from numerics (or with a well-established theoretical value).

In practice, the choice of which quantity to use will depend on convenience. If you are already set up to measure the temperature dependence of specific heat, you might do that. You might also measure $T_c$, or the magnetic field required to achieve total saturation. The choice will also be influenced by the parameter regime (temperature, field, pressure) in which the comparison to the theoretical model is the most relevant.

Here are a couple examples:

• In this neutron scattering study, they determined $J$ by comparing the magnon excitation dispersion (measured with neutron scattering) to the theoretical predictions. Dally et al. Phys. Rev. Lett. 124, 197203 (2020)
• In this field-polarized study, they used the magnetic susceptibility fit to the (previously documented) Bonner-Fisher result. Blanc, Trinh et al. Nat. Phys. 14, 273 (2018)