  The scattering of dark matter (DM) particles with sub-GeV masses off nuclei
is difficult to detect using liquid xenon-based DM search instruments because
the energy transfer during nuclear recoils is smaller than the typical detector
threshold. However, the tree-level DM-nucleus scattering diagram can be
accompanied by simultaneous emission of a Bremsstrahlung photon or a so-called
"Migdal" electron. These provide an electron recoil component to the
experimental signature at higher energies than the corresponding nuclear
recoil. The presence of this signature allows liquid xenon detectors to use
both the scintillation and the ionization signals in the analysis where the
nuclear recoil signal would not be otherwise visible. We report constraints on
spin-independent DM-nucleon scattering for DM particles with masses of 0.4-5
GeV/c$^2$ using 1.4$\times10^4$ kg$\cdot$day of search exposure from the 2013
data from the Large Underground Xenon (LUX) experiment for four different
classes of mediators. This analysis extends the reach of liquid xenon-based DM
search instruments to lower DM masses than has been achieved previously.
