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