Two Grid Points Promote The Nature Of The Bose-Habad Model

An extended Bose-Hubbard model is constructed by the local E₂ generators built from boson operators.Similar to the original Bose-Hubbard model,the extended model can be used to describe the quantum phase transition from the super-fluid phase to the Mott insulator phase.Some notable features of the quasi-exact solutions to the model are discussed,in which the ground state of the model is always included in the quasi-exact solutions.The ground state solution of the two-site Bose-Hubbard model with the number of particles N～105 can be quickly obtained by using the Mathematica code.Some excited levels of the model in 1D without local potential and on-site disorder for M = 2 and N = 10,N = 80,N = 155 are obtained.The ground state probabilities to detect n particles on any site Pn,the entanglement η,the ground state expectation value of local particle number<ni>,the ground state local particle number fluctuation<（△ni）²>for N = 15,= 80,N = 300 as functions of the control parameter U/to are also analyzed.