A density-matrix renormalization group Study of one-dimensional   incommensurate quantum Frenkel-Kontorova model
  In this paper, the one-dimensional incommensurate quantum Frenkel-Kontorova model is investigate by a density-matrix renormalization group algorithm. Special attention is given to the entanglement and the ground state energy. The energy gap between the ground state and the first excited is also calculated. From all the numerical results, we have observed an obvious property changes from the pinned state to the sliding one as the quantum fluctuation is increased. But no expected quantum critical point can be justified by the present data. 
Yongjun Ma, Jiaxiang Wang, Xinye Xu, Qi Wei, Sabre Kais
cond-mat.stat-mech quant-ph