  Nematic systems are two component superconductors that break rotational
symmetry, but exhibit a mixed symmetry that couples spatial rotations and phase
difference rotations. We show that a consequence of this induced spatial
anisotropy is mixed normal modes, that is the linear response to a small
perturbation of the system about its ground state, generally couples magnetic
and condensate degrees of freedom. We will study the effect of mode mixing on
the magnetic response of a nematic system as the strength of applied field is
increased. In general we show that the coupled modes generate magnetic field
perpendicular to the applied field, causing the magnetic response to
spontaneously twist direction. We will study this for the Meissner effect with
weak fields and also for stronger applied fields, which produce a mixture of
Skyrmions and composite vortices, forming orientation dependent bound states.
We will also calculate the anisotropies of the resulting first and second
critical fields $H_{c_1}$ and $H_{c_2}$. The Skyrmion lattices for $H_{c_1}
\leq H \leq H_{c_2}$ in nematic superconductors are shown to be structurally
complicated, in contrast to the triangular or square vortex lattices in
conventional superconductors. For low fields the magnetic response of the
system involves a loosely bound collection of parallel Skyrmion chains. As the
external field is increased the chains attract one another, causing a
transition where the unit cell becomes triangular for high applied fields. This
unique Skyrmion lattice and the magnetic twisting are clear indicators that
could be used experimentally to identify materials that exhibit nematic
superconductivity. To obtain these results we develop and present a novel
method to find the unit cell of a vortex lattice that can be applied to other
kinds of superconducting systems.
