Quantum phase transition by cyclic four-spin exchange interaction for S=1/2 two-leg spin ladder

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We investigate an S=1/2 two-leg spin ladder with a cyclic four-spin exchange interaction whose interaction constant is denoted by J₄, by using the density matrix renormalization group method. The interchain and the intrachain interaction constant are denoted by Jrung and J leg, respectively and assumed to be antiferromagnetic. It turns out that a spin gap between the singlet (Stotᶻ=0) and the triplet (Stotᶻ=1) states vanishes at J₄/Jleg ≃ 0.3 for Jrung=Jleg. This result is in contrast with the fact that the S=1/2 antiferromagnetic Heisenberg ladder, that is the case of Jrung ≠ Jleg, J₄=0, has a spin gap for all nonzero value of interchain interaction J rung>0. We find a larger value of the correlation length for the spin-pair correlation function than a linear size L of the system at J₄/J leg=0.3 and Jrung=Jleg: the correlation length ξ is about 204 times of the lattice constant for L=84 for these values of interactions. We also find that the string correlation function decays rather algebraically than exponentially at J₄/Jleg=0.3 and Jrung=Jleg. These results suggest that there is a quantum phase transition at J₄/Jleg ≃ 0.3 for Jrung=Jleg. We estimate a phase boundary where the spin gap vanishes in a J₄/Jleg-Jrung/Jleg plane and obtain a consistent result with that by a perturbation theory for J rung/Jleg>1.