# 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 $10^{-4}J$.