Untangling Magnetic Complexity in Protoplanetary Disks with the Zeeman
  Effect
  With the recent advent of circular polarization capabilities at the Atacama
Large Millimeter/submillimeter Array (ALMA), Zeeman effect measurements of
spectral lines are now possible as a means to directly probe line-of-sight
magnetic fields in protoplanetary disks (PPDs). We present a modeling study
that aims to guide physical interpretation of these anticipated observations.
Using a fiducial density structure based on a typical ringed disk, we simulate
line emission for the hyperfine components of the CN $J = 1-0$ transition with
the POLARIS radiative transfer code. Since the expected magnetic field and
typical CN distribution in PPDs remain largely unconstrained, we produce models
with several different configurations. Corresponding integrated Stokes $I$ and
$V$ profiles and 0.4 km/s resolution, 1'' beam convolved channel maps are
presented. We demonstrate that the emission signatures from toroidally
dominated magnetic fields are distinguishable from vertically dominated
magnetic field based on channel map morphology. Due to line-of-sight and beam
cancellation effects, disks with toroidal $\boldsymbol{B}$-field configurations
result in significantly diminished Stokes $V$ emission. Complex magnetic fields
therefore render the traditionally used method for inferring line-of-sight
magnetic field strengths (i.e., fitting the derivative of the Stokes $I$ to the
Stokes $V$ profile) ambiguous, since a given intrinsic field strength can yield
a variety of Stokes $V$ amplitudes depending on the magnetic field geometry. In
addition, gas gaps can create structure in the integrated Stokes $V$ profile
that might mimic magnetic substructure. This method should therefore be applied
with caution in PPD environments, and can only confidently be used as a measure
of magnetic field strength if the disk's magnetic field configuration is well
understood.
