  The merger of a neutron star binary may result in the formation of a
rapidly-spinning magnetar. The magnetar can potentially survive for seconds or
longer as a supramassive neutron star before collapsing to a black hole if,
indeed, it collapses at all. During this process, a fraction of the magnetar's
rotational energy of ~10^53 erg is transferred via magnetic spin-down to the
surrounding ejecta. The resulting interaction between the ejecta and the
surrounding circumburst medium powers a >year-long synchrotron radio transient.
We present a search for radio emission with the Very Large Array following nine
short-duration gamma-ray bursts (GRBs) at rest-frame times of ~1.3-7.6 years
after the bursts, focusing on those events which exhibit early-time excess
X-ray emission that may signify the presence of magnetars. We place upper
limits of <18-32 microJy on the 6.0 GHz radio emission, corresponding to
spectral luminosities of <(0.05-8.3)x10^39 erg s^-1. Comparing these limits to
the predicted radio emission from a long-lived remnant and incorporating
measurements of the circumburst densities from broad-band modeling of short GRB
afterglows, we rule out a stable magnetar with an energy of 10^53 erg for half
of the events in our sample. A supramassive remnant that injects a lower
rotational energy of 10^52 erg is ruled out for a single event, GRB 050724A.
This study represents the deepest and most extensive search for long-term radio
emission following short GRBs to date, and thus the most stringent limits
placed on the physical properties of magnetars associated with short GRBs from
radio observations.
