  We investigate the low-energy collective charge excitations (plasmons,
holons) in metallic atomic wires deposited on semiconducting substrates. These
systems are described by two-dimensional correlated models representing
strongly anisotropic lattices or weakly coupled chains. Well-established
theoretical approaches and results are used to study their properties: random
phase approximation for anisotropic Fermi liquids and bosonization for coupled
Tomonaga-Luttinger liquids as well as Bethe Ansatz and density-matrix
renormalization group methods for ladder models. We show that the Fermi and
Tomonaga-Luttinger liquid theories predict the same qualitative behavior for
the dispersion of excitations at long wave lengths. Moreover, their scaling
depends on the choice of the effective electron-electron interaction but does
not characterize the dimensionality of the metallic state. Our results also
suggest that such anisotropic correlated systems can exhibit two-dimensional
dispersions due to the coupling between wires but remain quasi-one-dimensional
strongly anisotropic conductors or retain typical features of
Tomonaga-Luttinger liquids such as the power-law behaviour of the density of
states at the Fermi energy. Thus it is possible that atomic wire materials such
as Au/Ge(100) exhibit a mixture of features associated with one and two
dimensional metals.
