  In this work, we extend the analysis of interacting bosons at 2D-1D
dimensional crossover for finite size and temperature by using field-theory
approach (bosonization) and quantum Monte Carlo simulations. Stemming from the
fact that finite size low-dimensional systems are allowed only to have
quasi-ordered phase, we consider the self-consistent harmonic approximation to
compute the fraction of quasi-condensate and its scaling with the system size.
It allows us to understand the important role played by finite size and
temperature across the dimensional crossover for deciding the condensate
nature. Furthermore, we consider a mean-field approximation to compute the
finite size effect on the crossover temperature for both weak and strong
interaction. All the physical quantities we discuss here provide essential
information for quantum gas at dimensional crossover and are directly
detectable in cold atom experiments
