  In this paper, we construct the Rabinowitz-Floer homology for the coupled
Dirac system \begin{equation*} \left\{ \begin{aligned} Du=\frac{\partial
H}{\partial v}(x,u,v)\hspace{4mm} {\rm on} \hspace{2mm}M,\\ Dv=\frac{\partial
H}{\partial u}(x,u,v)\hspace{4mm} {\rm on} \hspace{2mm}M, \end{aligned} \right.
\end{equation*} where $M$ is an $n$-dimensional compact Riemannian spin
manifold, $D$ is the Dirac operator on $M$, and $H:\Sigma M\oplus \Sigma M\to
\mathbb{R}$ is a real valued superquadratic function of class $C^1$ with
subcritical growth rates. Solutions of this system can be obtained from the
critical points of a Rabinowitz-Floer functional on a product space of suitable
fractional Sobolev spaces. In particular, we consider the $S^1$-equivariant $H$
that includes a nonlinearity of the form $$
H(x,u,v)=f(x)\frac{|u|^{p+1}}{p+1}+g(x)\frac{|v|^{q+1}}{q+1}, $$ where $f(x)$
and $g(x)$ are strictly positive continuous functions on $M$, and $p>1,q>1$
satisfy $$ \frac{1}{p+1}+\frac{1}{q+1}>\frac{n-1}{n}. $$ We establish the
existence of a nontrivial solution by computing the Rabinowitz-Floer homology
in the Morse-Bott situation.
