Different types of the dimensional crossover in the quasi-one-dimensional spinless fermion systems

It is known that many-body correlations qualitatively modify the properties of a one-dimensional metal. However, for a quasi-one-dimensional metal these correlations are suppressed, at least partially. We study conditions under which the one-dimensional effects significantly influence the dimensional crossover of a quasi-one-dimensional metal. It is proved (i) that even a system with very high anisotropy of the single-particle hopping might behave on both sides of the crossover as an ordinary weakly non-ideal Fermi gas. Further, (ii) to demonstrate well-developed signatures of one-dimensional correlations the system must have extremely (exponentially) high anisotropy. Between cases (i) and (ii) an intermediate regime lies: (iii) the one-dimensional phenomena affect the two-particle susceptibilities, but do not reveal themselves in single-particle quantities. Unlike the normal state properties, (iv) the ordering transition is always very sensitive to the anisotropy: the mean field theory quickly becomes invalid as the anisotropy increases. An expression for the transition temperature is derived. The attributes (i-iv) are used to classify the weakly interacting quasi-one-dimensional fermion systems.