  We measure the transmission phase of a quantum point contact (QPC) at a low
carrier density in which electron interaction is expected to play an important
role and anomalous behaviors are observed. In the first conductance plateau,
the transmission phase shifts monotonically as the carrier density is decreased
by the gate voltage. When the conductance starts to decrease, in what is often
called the 0.7 regime, the phase exhibits an anomalous increase compared with
the noninteracting model. The observation implies an increase in the wave
vector as the carrier density is decreased, suggesting a transition to a
spin-incoherent Luttinger liquid.
