LAUSR

We construct a four-leg spin-1/2 $t$-$J$ type model to simulate a doped two-leg spin-1 antiferromagnetic Heisenberg ladder. Employing renormalized mean-field theory with simple Gutzwiller factors, we obtain three degenerate superconducting states with different pairing symmetry. Through improving the Gutzwiller factors, we find that the state C with inter-layer modified $d_{y^2-z^2}$-wave pairing has the lowest energy in a large doping range. Besides, we use the density matrix renormalization group method to solve the model. The negative binding energy reveals the pairing tendency, and the pair-pair correlation functions exhibit a slowly decaying behavior on certain types of bonds. From the pair correlations, we confirm the inter-layer modified $d_{y^2-z^2}$-wave superconducting state as the ground state of the model.