  Solar-type stars are born with relatively rapid rotation and strong magnetic
fields. Through a process known as magnetic braking, the rotation slows over
time as stellar winds gradually remove angular momentum from the system. The
rate of angular momentum loss depends sensitively on the magnetic morphology,
with the dipole field exerting the largest torque on the star. Recent
observations suggest that the efficiency of magnetic braking may decrease
dramatically in stars near the middle of their main-sequence lifetimes. One
hypothesis to explain this reduction in efficiency is a shift in magnetic
morphology from predominantly larger to smaller spatial scales. We aim to test
this hypothesis with spectropolarimetric measurements of two stars that sample
chromospheric activity levels on opposite sides of the proposed magnetic
transition. As predicted, the more active star (HD 100180) exhibits a
significant circular polarization signature due to a non-axisymmetric
large-scale magnetic field, while the less active star (HD 143761) shows no
significant signal. We identify analogs of the two stars among a sample of
well-characterized Kepler targets, and we predict that the asteroseismic age of
HD 143761 from future TESS observations will substantially exceed the age
expected from gyrochronology. We conclude that a shift in magnetic morphology
likely contributes to the loss of magnetic braking in middle-aged stars, which
appears to coincide with the shutdown of their global dynamos.
