  Most quantum point contacts (QPCs) fabricated in high-mobility 2D electron
gases show a zero-bias conductance peak near pinchoff, but the origin of this
peak remains a mystery. Previous experiments have primarily focused on the
zero-bias peak at moderate conductance, in the range (1-2)e^2/h. Here,
measurements are presented of zero-bias peaks that persist down to
10^{-4}e^2/h. Magnetic field and temperature dependencies of the zero-bias peak
in the low-conductance limit are qualitatively different from the analogous
phenomenology at higher conductance, with implications for existing theoretical
models of transport in low-density QPCs.
