# Magetic Properties

Below Neel temperature, $T_N$, magnetic moments residing on the Cu$^{2+}$ ions in the CuO2 planes of cuprates order antiferromagnetically.

Qualitatively, this physics is captured by the Heisenberg model, which describes interaction between idealized spin-1/2 Cu ions. The basic model is $H = \sum_{i,j} J_{ij}{\bf S}_i\cdot {\bf S}_j$ ^{[1]}. The strongest interaction is between the nearest-neighbor sites in a CuO2 plane, which has typical values between 100 and 150 meV. Naively, one would expect that the energy scale of $J$ would translate into the Neel temperature above 1000K. However, in real materials, the typical range of Neel temperatures is between 250 and 400 K.

In the idealized case of 2D nearest neighbor Heisenberg model, the moments only order at zero temperature, with the value of the ordered moment $\langle S\rangle = 0.3$, down from the nominal value of $S = 1/2$ ^{[2]}. Therefore, the actual value of Neel temperature depends critically on the deviations from the isotropic two-dimensional Heisenberg model, that is, on the value and type of spin orbit interaction, and on the interlayer coupling. Indeed, experimentally, there is a lot of variability in $T_N/J$, as can be seen from attached table. Given the Lande g-factor 2.1 for Cu, the classical value of the magnetic ordered moment would be about 1.1 $\mu_B$, which in the case of the basic Heisnberg model would be brought down to about 0.66 $\mu_B$. The fact that experimentally observed values are consistently lower indicates that the pure-spin description provided by the Heisenberg model may not be sufficient, and thus models of the (multi-orbital) Hubbard type may be needed to account for the extra reduction of the ordered moment ^{[3]}.

Within the realm of pure spin models, the parameters can be determined by fitting the experimental data on spin wave dispersion, which can be obtained from inelastic neutron scattering. This allows to determine longer-range exchange interactions, as well multi (more than two) spin interaction strengths. For instance, for La$_2$CuO$_4$, introducing a four-spin cyclic exchange tern of about 60 meV leads to a good fit ^{[4]}.

Spin orbit interaction is another ingredient that needs to be included in order to account for the preferential orientation of magnetic moments relative to the crystalline axes. For spin 1/2, there can be no single ion anisotropy terms; therefore it can only affect the inter-spin interaction, manifesting as anisotropic exchange or Dzyaloshinsky-Moriya.