Effects of a space modulation on the behavior of a 1D alternating   Heisenberg spin-1/2 model
  The effects of a magnetic field ($h$) and a space modulation ($\delta$) on the magnetic properties of a one dimensional antiferromagnetic-ferromagnetic Heisenberg spin-1/2 model have been studied by means of numerical exact diagonalization of finite size systems, nonlinear sigma model and bosonization approach. The space modulation is considered on the antiferromagnetic couplings. At $\delta=0$, the model is mapped to a gapless L\"{u}ttinger liquid phase by increasing the magnetic field. However, the space modulation induces a new gap in the spectrum of the system and the system experiences different quantum phases which are separated by four critical fields. By opening the new gap a magnetization plateau appears at ${1/2}M_{sat}$. The effects of the space modulation are reflected in the emergence of a plateau in other physical functions such as the F-dimer and the bond dimer order parameters, and the pair-wise entanglement. 
Saeed Mahdavifar and Jahanfar Abouie
cond-mat.str-el cond-mat.stat-mech quant-ph