Dimensional Crossover in the Bose–Einstein Condensation Confined to Anisotropic Three-Dimensional Lattices

Abstract

AbstractThe Bose–Einstein condensation (BEC) in three-dimensional (3D) anisotropic lattices is studied. We present theoretical results for the critical temperature for BEC, chemical potential, condensate fraction and relevant thermodynamic quantities like: internal energy, entropy, specific heat and compressibility as a function of anisotropy parameter being the ratio of the nearest-neighbor in-plane ($$t_\parallel$$ t ‖ ) and out-of-plane ($$t_\perp$$ t ⊥ ) hopping amplitudes. In particular, considered scenarios include weakly coupled two-dimensional (2D) planes ($$t_\perp /t_\parallel \ll 1$$ t ⊥ / t ‖ ≪ 1 , relevant for layered structures) as well as a rod-like geometry of interacting one-dimensional (1D) chains ($$t_\parallel /t_\perp \ll 1$$ t ‖ / t ⊥ ≪ 1 ). The impact of the dimensional crossover as the system is tuned away from a set of disconnected 2D layers, or traverses from a set of separate 1D chains to a regime where a fully isotropic 3D structure emerges is elucidated. Both numerical and analytic approaches are employed, (the latter in a form of series expansions involving $$t_\parallel ,t_\perp$$ t ‖ , t ⊥ amplitudes) for internal energy, entropy, specific heat and isothermal compressibility. The theoretical outcome of the present study may be of interest to a number of scenarios in solid-state physics, where the relevant quasi-particles are bosonic-like, as well as might be applicable to the physics of cold bosons loaded in artificially engineered 3D optical lattices.