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

This question is related to other one here in the MatterModelingSE:

Here J is the exchange interaction parameters between two nearest-neighbor spins using in Heisenberg and Ising hamiltonians to study magnetic systems:

$$$$\tag{Heisenberg} \hat{H}_H=-\sum_{\langle i j\rangle}J\hat{S}_i\hat{S}_j$$$$

$$$$\tag{Ising} \hat{H}_I=-\sum_{\langle ij\rangle}J\hat{S}_i^z\hat{S}_j^z$$$$

Is there an experimental (and measurable) observable that can be related to the J parameter?

• Is the Hund's J the same as the Heisenberg/Ising J? Jul 22, 2020 at 1:39

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:

• +1, and agreed. If you have a Heisenberg or Ising model, the only two energy scales in the problem are $J$ and the temperature $T$ (assuming spin size $S$, and number of lattice sites $N$ are treated as parameters), so most measurable quantities will have some $J$-dependence. In practice, the two you've listed are quite commonly used, and works also for more complicated Hamiltonians. The one caveat I'd add is that susceptibility is often fit to a high-temperature expansion, whereas neutron scattering fits are usually done at low temperatures, which can sometimes lead to discrepancies. Jul 24, 2020 at 14:01