  Quasi-one-dimensional (Q1D) systems, i.e., three- and two-dimensional (3D/2D)
arrays composed of weakly coupled one-dimensional lattices of interacting
quantum particles, exhibit rich and fascinating physics. They are studied
across various areas of condensed matter and ultracold atomic lattice-gas
physics, and are often marked by dimensional crossover as the coupling between
one-dimensional systems is increased or temperature decreased, i.e., the Q1D
system goes from appearing largely 1D to largely 3D. Phase transitions
occurring along the crossover can strongly enhance this effect. Understanding
these crossovers and associated phase transitions can be challenging due to the
very different elementary excitations of 1D systems compared to
higher-dimensional ones. In the present work, we combine numerical matrix
product state (MPS) methods with mean-field (MF) theory to study paradigmatic
cases of dimensional crossovers and the associated phase transitions in systems
of both hard-core and soft-core lattice bosons, with relevance to both
condensed matter physics and ultracold atomic gases. We show that the
superfluid-to-insulator transition is a first order one, as opposed to the
isotropic cases and calculate transition temperatures for the superfluid
states, finding excellent agreement with analytical theory. At the same time,
our MPS+MF approach keeps functioning well where the current analytical
framework cannot be applied. We further confirm the qualitative and
quantitative reliability of our approach by comparison to exact quantum Monte
Carlo calculations for the full 3D arrays.
