Modification of the Bose-Hubbard model parameters in an optical lattice by two-particle Wannier functions

In systems with periodic potential fields, building relatively local Wannier functions can significantly simplify the Hamiltonian and enhance our understanding of the system's ground state and dynamic properties. In this work, we improve the current method of building the Wannier functions of ultracold atomic systems, including the case in the presence or absence of interactions. In noninteracting systems, we propose a method to directly obtain the real-valued maximally localized Wannier functions (MLWFs) by using real-valued eigenstates, and verify the effectiveness of this method in a two-dimensional (2D) degenerate system. In interacting systems, we obtain the effect of high-energy bands on the lowest-energy band by using the accurate calculation results of the two-particle system. In the two-particle system, we consider the effect of the entanglement between the particles and obtain the optimal two-particle Wannier functions. These Wannier functions are then further utilized to obtain the parameters of the extended Bose-Hubbard model. The effectiveness of the method is verified by taking a one-dimensional (1D) system with contact interaction as an example. In the three-particle and four-particle systems, compared calculation results with the original system and the unmodified two-band Bose-Hubbard model, we find that the effective Hamiltonian is more accurate than the unmodified two-band model. This verifies the effectiveness of our method, and the parameters obtained can reflect the original system well, which provides an effective method for accurate modeling of interacting systems.