$\newcommand{\lco}{{\rm La}_2{\rm CuO}_4}$
$\newcommand{\bot}{\perp}$
$\newcommand{\|}{\parallel}$
$\newcommand{\ybco}{\rm YBa_2Cu_3O_{6+x}}$

Measurement of magnetic excitations in the insulating state allows accurate calibration of effective Heisenberg-like models of undoped cuprates.

The starting point for considering magnetic interactions in the cuprates
is the Heisenberg hamiltonian:
\begin{equation}
 H = J \sum_{\langle i,j\rangle} {\bf S}_i\cdot {\bf S_j},
\end{equation}
where $\langle i,j\rangle$ denotes all nearest-neighbor pairs,
each included once. Spin-wave theory
can be applied to the Heisenberg Hamiltonian to calculate the dispersion
of spin fluctuations about ${\bf Q}_{\rm AF}$ [[oguchi]].  At low energies the spin
waves disperse linearly with ${\bf q} = {\bf Q} - {\bf Q}_{\rm AF}$ (see
Figure), having a velocity $c = \sqrt{8}SZ_c Ja/\hbar$,
where $Z_c\approx 1.18$ [[singh]] is a quantum-renormalization
factor.  Thus, by measuring the spin-wave velocity, one can determine
$J$.  

Spin-wave measurements have been performed for a number of cuprates, and
some results for $J$ are listed in Table of [Magnetic Properties](/nodes/124) .  Complementary measurements of $J$ can be obtained by two-magnon Raman scattering
[[lyons]].  To calculate the values of $J$ from spectroscopically
determined parameters, one must consider at least a 3-band Hubbard model
[[emery]].  Recent *ab initio* cluster calculations
[[abin]] have been able to achieve reasonable agreement with
experiment.

*** 

To describe the experimental dispersion curves in greater detail, particularly at high excitation energies, one must add more terms to the spin Hamiltonian. 
For example, in a La$_2$CuO$_4$, the observed
dispersion along the zone boundary (attached Figure), between $(\frac12,0)$ and
$(\frac34,\frac14)$, is not expected based on the simple Heisenberg model.  To account for it, one can include the additional terms that, e.g., appear when the
perturbation expansion for the single-band Hubbard model is extended to
fourth order [[coldea]].  The most important new term involves 4-spin cyclic
exchange about a plaquette of four Cu sites [[cyclic]]. 
The data can be fit quite well at 10 K with $J=146(4)$ meV and a cyclic exchange energy
$J_c=61(8)$ meV.

Alternatively, one can try to fit the dispersion by including longer range magnetic exchange interactions, such as the next-nearest neighbor interaction $J'$. 
It turns out, that fitting the measured dispersion with only $J$ and $J'$
requires that $J'$ correspond to a ferromagnetic interaction, which is inconsistent with $J'$ originating from a longer range hopping within the single band Hubbard model.

***

At low energies, there are other terms that need to be considered.  There
need to be anisotropies, with associated spin-wave gaps, in order to fix
the spin direction; however, an atom with $S=\frac12$ cannot have
single-ion anisotropy.  Instead, the anisotropy is associated with the
nearest-neighbor superexchange interaction.  Consider a pair of
nearest-neighbor spins, ${\bf S}_i$ and ${\bf S}_j$, within a CuO$_2$
plane, with each site having tetragonal symmetry.  The Heisenberg
Hamiltonian for this pair can be written
$$
  H_{\rm pair} = J_\| S_i^\| S_j^\| + J_\bot S_i^\bot S_j^\bot +
     J_z S_i^z S_j^z,
$$
where $\|$ and $\bot$ denote directions parallel and perpendicular to the
bond within the plane, and $z$ is the out-of-plane direction.  To discuss the anisotropies, it is convenient to define
the quantities $\Delta J \equiv J_{\rm av} - J_z$, where $J_{\rm av}
\equiv (J_\| + J_\bot)/2$, and $\delta J_{\rm in} \equiv J_\| - J_\bot$
[[yild]].  For the cuprates, $J_{\rm av} \gg \Delta J > \delta
J_{\rm in} > 0$.  The out-of-plane anisotropy, $\alpha_{\rm XY} = \Delta
J/J_{\rm av}$, causes the spins to lie, on average, in the $x$-$y$ plane,
and results in a spin-wave gap for out-of-plane fluctuations.  The
in-plane anisotropy
$\delta J_{\rm in}/J_{\rm av}$, contributing through the quantum
zero-point energy [[yilda]], tends to favor alignment of the
spins parallel to a bond direction, and causes the in-plane spin-wave
mode to have a gap.  The effective coupling between planes (which can
involve contributions from several interactions [[yilda]] leads to
(weak) dispersion along $Q_z$.

For stoichiometric $\lco$, the out-of-plane spin gap is 5.5(5) meV,
corresponding to $\alpha_{\rm XY} = 1.5\times 10^{-4}$ [[keimer]]. The
in-plane gap of 2.8(5) meV has a contribution from anisotropic exchange
of the Dzyaloshinsky-Moriya type [[dm]], as well as from
$\delta J_{\rm in}$.  No dispersion along $Q_z$ has been reported.

For antiferromagnetic $\ybco$, an out-of-plane gap of about 5 meV has been
observed [[ybco]], indicating an easy-plane anisotropy
similar to that in $\lco$. No in-plane gap has been resolved; however,
the in-plane mode shows a dispersion of about 3 meV along $Q_z$ [[ybco]].  The latter dispersion is controlled by the effective exchange between Cu moments in neighboring bilayers through the nonmagnetic Cu(1) sites, which is on the order of $10^{-4}J$.