Gapped spin liquid states in a one-dimensional Hubbard model with   antiferromagnetic exchange interaction
  We study the phase diagram of a one-dimensional extended Hubbard model with antiferromagnetic exchange interaction analytically and numerically. The bosonization and transfer-matrix renormalization group methods are used in the corresponding coupling regimes. At half-filling, the system is a Mott insulator with a finite spin excitation gap if the on-site Coulomb repulsion is fairly smaller than the antiferromagnetic exchange J. This Mott-insulator is characterized by the bond-charge-density-wave order or spontaneously dimerization. In the weak-coupling regime where the spin-charge separation holds approximately, the critical point separating the gapless and gapped spin liquid phases is U_c\sim J/2. However, as J increases, the spin-charge couplings become important and the critical point U_c is significantly suppressed and eventually tends to zero as J\to \infty. Away from half-filling, the charge gap completely collapses but the spin gap persists. 
Jianhui Dai, Xiaoyong Feng, Tao Xiang, Yue Yu