We demonstrate that dwarf galaxies (10^7 < M_stellar < 10^9 Msun) with no
active star formation are extremely rare (<0.06%) in the field. Our sample is
based on the NASA-Sloan Atlas which is a re-analysis of the Sloan Digital Sky
Survey Data Release 8. We examine the relative number of quenched versus star
forming dwarf galaxies, defining quenched galaxies as having no Halpha emission
(EW_Halpha < 2 AA) and a strong 4000AA-break. The fraction of quenched dwarf
galaxies decreases rapidly with increasing distance from a massive host,
leveling off for distances beyond 1.5 Mpc. We define galaxies beyond 1.5 Mpc of
a massive host galaxy to be in the field. We demonstrate that there is a
stellar mass threshold of M_stellar < 1.0x10^9 Msun below which quenched
galaxies do not exist in the field. Below this threshold, we find that none of
the 2951 field dwarf galaxies are quenched; all field dwarf galaxies show
evidence for recent star formation. Correcting for volume effects, this
corresponds to a 1-sigma upper limit on the quenched fraction of 0.06%. In more
dense environments, quenched galaxies account for 23% of the dwarf population
over the same stellar mass range. The majority of quenched dwarf galaxies
(often classified as dwarf elliptical galaxies) are within 2 virial radii of a
massive galaxy, and only a few percent of quenched dwarf galaxies exist beyond
4 virial radii. Thus, for galaxies with stellar mass less than 1.0x10^9 Msun,
ending star-formation requires the presence of a more massive neighbor,
providing a stringent constraint on models of star formation feedback.Comment: 9 pages, 6 figures, accepted to Ap