Magnetic excitations
$\newcommand{\lco}{{\rm La}_2{\rm CuO}_4}$
$\newcommand{\bot}{\perp}$
$\newcommand{\|}{\parallel}$
$\newcommand{\ybco}{\rm YBa_2Cu_3O_{6+x}}$
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 ${\bf Q}_{\rm AF}$ [1]. 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 $Z_c\approx 1.18$ [2] 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 . Complementary measurements of $J$ can be obtained by two-magnon Raman scattering
[3]. To calculate the values of $J$ from spectroscopically
determined parameters, one must consider at least a 3-band Hubbard model
[4]. Recent ab initio cluster calculations
[5] 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 [6]. The most important new term involves 4-spin cyclic
exchange about a plaquette of four Cu sites [7].
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 $\delta J_{\rm in} \equiv J_\| - J_\bot$
[8]. 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 [9], 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 [9] leads to
(weak) dispersion along $Q_z$.
For stoichiometric $\lco$, the out-of-plane spin gap is 5.5(5) meV,
corresponding to $\alpha_{\rm XY} = 1.5\times 10^{-4}$ [10]. The
in-plane gap of 2.8(5) meV has a contribution from anisotropic exchange
of the Dzyaloshinsky-Moriya type [11], 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 [12], 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 $Q_z$ [12]. 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 $10^{-4}J$.