  The Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction has been extensively
explored in isotropic Dirac systems with linear dispersion, which typically
follows an exponent decaying rate with the impurity distance $R$, i.e.,
$J\propto 1/R^d$ ($1/R^{2d-1}$) in $d$-dimensional systems at finite (zero)
Fermi energy. This fast decay makes it rather difficult to be detected and
limits its application in spintronics. Here, we theoretically investigate the
influence of anisotropic dispersion on the RKKY interaction, and find that the
introduction of non-relativistic dispersion in semi-Dirac semimetals (S-DSMs)
can significantly prolong the decay of the RKKY interaction and can remarkably
enhance the Dzyaloshinskii-Moriya interaction around the relativistic
direction. The underlying physics is attributed to the highly increased density
of states in the linear-momentum direction as a result of the interplay of
relativistic and non-relativistic electrons. Furthermore, we propose a general
formula to determine the decaying rate of the RKKY interaction, extending the
typical formula for isotropic DSMs. Our results suggest that the S-DSM
materials are a powerful platform to detect and control the magnetic exchange
interaction, superior to extensively adopted isotropic Dirac systems.
