We study ground-state quantum entanglement in the one-dimensional Bose-Hubbard model in the presence of a harmonic trap. We focus on two transitions that occur upon increasing the characteristic particle density: the formation of a Mott-insulating domain with site occupation one at the center of the trap (lower transition) and the emergence of a superfluid domain at the center of the Mott-insulating one (upper transition). These transitions generate discontinuities in derivatives of the total energy and have been characterized by local (nonextensive) order parameters, so we refer to them as local quantum phase transitions. We show that a second derivative of the total energy is continuous with a kink at the lower transition, and that it is discontinuous at the upper transition. We also show that bipartite entanglement entropies are order parameters for those local quantum phase transitions. We use the density-matrix renormalization group and show that the transition points extracted from entanglement measures agree with the predictions of the local density approximation in the thermodynamic limit. We discuss how to determine the transition points from results in small systems, such as the ones realized in recent optical lattice experiments that measured the second-order Renyi entanglement entropy.
The effect of interactions on Bose-Einstein condensation in a two-dimensional harmonic trap
研究了准二维简谐势阱中的稀薄玻色气体与具有固定耦合常数的排斥性两体零程势的相互作用。使用T-Fermi方法，当温度降低时，它仍保持在相同的非凝聚相。它的密度分布和能量与服从Haldron分数不相容统计的理想气体的密度分布和能量相同。PACS：~03.75.Fi，05.30.Jp，67.40.Db，05.30.- D
