  We develop a general theory for how an exciton in an atomically thin
transition metal dichacogenide (TMD) monolayer couples to spin and charge
correlations in an adjacent moire lattice created by a TMD bi-layer. Virtual
tunneling of charge carriers, assumed for concreteness to be holes, between the
moire lattice and the monolayer combined with the presence of bound
hole-exciton states, i.e. trions, give rise to an effective interaction between
the moire holes and the exciton. In addition to the Umklapp scattering, we show
that this interaction is spin-dependent and therefore couples the exciton to
the spin correlations of the moire holes, which may be in- as well as
out-of-plane. We then use our theory to examine two specific examples where the
moire holes form in-plane ferromagnetic or anti-ferromagnetic order. In both
cases, the exciton creates spin waves in the moire lattice, which we analyse by
using a self-consistent Born approximation that includes such processes to
infinite order. We show that the competition between magnetic order and exciton
motion leads to the formation of a well-defined quasiparticle consisting of the
exciton surrounded by a cloud of magnetic frustration in the moire lattice
sites below. For the anti-ferromagnet, we furthermore demonstrate the presence
of the elusive geometric string excitations and discuss how they can be
observed via their smoking gun energy dependence on the spin-spin coupling,
which can be tuned by varying the twist angle of the moire bi-layer. All these
phenomena have clear signatures in the exciton spectrum, and as such our
results illustrate that excitons are promising probes providing optical access
to the spin correlations of new quantum phases predicted to exist in TMD
materials.
