Confinement properties of the 1+1 Schwinger model can be studied by computing the string tension between two charges. It is finite (vanishing) if the fermions are massive (massless), corresponding to the occurrence of confinement (screening). Motivated by the possibility of experimentally simulating the Schwinger model, we investigate here the robustness of its screened and confined phases. First, we analyze the effect of nearest-neighbor density-density interaction terms, which - in the absence of the gauge fields - give rise to the Thirring model. The resulting Schwinger-Thirring model (very often also referred to as the gauged Thirring model) is studied, also in presence of a topological θ-term, showing that the massless (massive) model remains screened (confined) and that there is deconfinement only for θ=±π in the massive case. Estimates of the parameters of the Schwinger-Thirring model are provided with a discussion of a possible experimental setup for its realization with ultracold atoms. The possibility that the gauge fields live in higher dimensions while the fermions remain in 1+1 is also considered. One may refer to this model as the pseudo-Schwinger-Thirring model. It is shown that the screening of external charges occurs for 2+1 and 3+1 gauge fields, exactly as it occurs in 1+1 dimensions, with a radical change of the long distance interaction induced by the gauge fields. The massive (massless) model continues to exhibit confinement (screening), signaling that it is the dimensionality of the matter fields, and not of the gauge fields, to determine confinement properties. A computation for the string tension is presented in perturbation theory. Our conclusion is that 1+1 models exhibiting confinement or screening - massless or massive, in the presence of a topological term or not - retain their main properties when the Thirring interaction is added or the gauge fields live in a higher dimension.
