XXZ spin-1/2 representation of a finite-U Bose-Hubbard chain at half-integer filling

Using a similarity Hamiltonian renormalization procedure, we determine an effective spin-1/2 representation of the Bose-Hubbard model at half-integer filling and at a finite on-site interaction energy U. By means of bosonization, we are able to recast the effective Hamiltonian as that of a spin-1/2 XXZ magnetic chain with pertinently renormalized coupling and anisotropy parameters. We use this mapping to provide analytical estimates of the correlation functions of the Bose-Hubbard model. We then compare such results with those based on DMRG numerical simulations of the Bose-Hubbard model for various values of U and for a number L of lattice sites as low as L ~ 30. We find an excellent agreement up to 10\% between the output of analytical and numerical computations, even for relatively small values of U. Our analysis implies that, also at finite U, the 1D Bose-Hubbard model with suitably chosen parameters may be seen as a quantum simulator of the XXZ chain.