Quantum phase transition in one-dimensional commensurate   Frenkel-Kontorova model
  In this paper, we have studied the one-dimensional commensurate quantum Frenkel-Kontorova model by a density-matrix renormalization group (DMRG) algorithm. The focus has been on its properties of the entanglement, the coordinate correlation, the ground state energy and the energy gap between the ground state and the first excited one. It is demonstrated that a quantum phase transition (QPT) between the pinned and the sliding phases takes place as the quantum fluctuation, measured by an effective Planck constant $\tilde\hbar$, increases to a threshold value $\tilde \hbar_c$. 
Yongjun Ma, Jiaxiang Wang, Xinye Xu, Qi Wei and Sabre Kais
cond-mat.stat-mech quant-ph