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Using the numerical diagonalization method, we examine the one-dimensionalt_1-t_2-J_1-J_2 model (zigzag chain t-J model) which represents an effectivemodel for metallic CuO double chain in the superconductorPr_2Ba_4Cu_7O_15-\delta. Based on the Tomonaga-Luttinger liquid theory, wecalculate the Luttinger-liquid parameter K_\rho as a function of electrondensity n. It is found that superconductivity is realized in parameter regioncorresponding to the experimental result. We show phase diagram of spin gap onthe t_2/|t_1|-n plane by analyzing the expectation value of twist-operatorZ_\sigma in the spin sector. The spin gap appears in the region with larget_2/|t_1|, where the phase boundary at half-filling is consistent with that ofthe known frustrated quantum spin system. The analysis also suggests that theestimated value of the spin gap reaches 100K in the realistic parameter regionof Pr_2Ba_4Cu_7O_15-delta.

Superconductivity and Spin gap in the zigzag chain t-J model simulating a CuO double chain in Pr_2Ba_4Cu_7O_15-delta