  The three-leg ladder has one odd-parity and two even-parity channels. At low
doping these behave quite differently. Numerical calculations for a t-J model
show that the initial phase upon hole doping has two components - a conducting
Luttinger liquid in the odd-parity channel, coexisting with an insulating (i.e.
undoped) spin liquid phase in the even-parity channels. This phase has a
partially truncated Fermi surface and violates the Luttinger theorem. This
coexistence of conducting fermionic and insulating paired bosonic degrees of
freedom is similar to the recent proposal of Geshkenbein, Ioffe, and Larkin for
the underdoped spin-gap normal phase of the cuprates. A mean field
approximation is derived which has many similarities to the numerical results.
One difference however is an induced hole pairing in the odd-parity channel at
arbitrary small dopings, similar to that proposed by Geshkenbein, Ioffe, and
Larkin for the two-dimensional case. At higher dopings, we propose that a
quantum phase transition will occur as holes enter the even-parity channels,
resulting in a Luther-Emery liquid with hole pairing with essentially d-wave
character. In the mean field approximation a crossover occurs which we
interpret as a reflection of this quantum phase transition deduced from the
numerical results.
