We establish a correspondence between the electric dipole matrix elements of
a polyatomic symmetric top molecule in a state with nonzero projection of the
total angular momentum on the symmetry axis of the molecule and the magnetic
dipole matrix elements of a magnetic dipole associated with an elemental spin
$F$. It is shown that this correspondence makes it possible to perform quantum
simulation of the single-particle spectrum and the dipole-dipole interactions
of magnetic dipoles in a static external magnetic field $\bf{B}$ with symmetric
top molecules subject to a static external electric field
$\bf{E}_{\mathrm{DC}}$. We further show that no such correspondence exists for
$^1\Sigma$ molecules in static fields, such as the alkali metal dimers. The
effective spin angular momentum of the simulated magnetic dipole corresponds to
the rotational angular momentum of the symmetric top molecule, and so quantum
simulation of arbitrarily large integer spins is possible. Further, taking the
molecule CH$_3$F as an example, we show that the characteristic dipole-dipole
interaction energies of the simulated magnetic dipole are a factor of 620, 600,
and 310 larger than for the highly magnetic atoms Chromium, Erbium, and
Dysprosium, respectively. We present several applications of our correspondence
for many-body physics, including long-range and anisotropic spin models with
arbitrary integer spin $S$ using symmetric top molecules in optical lattices,
quantum simulation of molecular magnets, and spontaneous demagnetization of
Bose-Einstein condensates due to dipole-dipole interactions. Our results are
expected to be relevant as cold symmetric top molecules reach quantum
degeneracy through Stark deceleration and opto-electrical cooling.Comment: 28 pages, 3 figures, submitted to the special issue of Annalen der
Physik on "Quantum Simulation"; v2: revised in response to referees' comment