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Confinement properties of the $1+1$ Schwinger model can bestudied bycomputing the string tension between two charges. It is finite (vanishing) ifthe fermions are massive (massless) corresponding to the occurrence ofconfinement (screening). Motivated by the possibility of experimentallysimulate the Schwinger model, we investigate here the robustness of itsscreened and confined phases. Firstly, we analyze the effect ofnearest-neighbour density-density interaction terms, which -- in the absence ofthe gauge fields -- give rise to the Thirring model. The resultingSchwinger-Thirring model is studied, also in presence of a topological $\theta$term, showing that the massless (massive) model remains screened (confined) andthat there is deconfinement only for $\theta=\pm\pi$ in the massive case.Estimates of the parameters of the Schwinger-Thirring model are provided with adiscussion of a possible experimental setup for its realization with ultracoldatoms. The possibility that the gauge fields live in higher dimensions whilethe fermions remain in $1+1$ is also considered. One may refer to this model asthe Pseudo-Schwinger-Thirring model. It is shown that the screening of externalcharges 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 bythe gauge fields. The massive (massless) model continues to exhibit confinement(screening), signalling that it is the dimensionality of the matter fields, andnot of the gauge fields to determine confinement properties. A computation forthe string tension is presented in perturbation theory. Our conclusion is that$1+1$ models exhibiting confinement or screening -- massless or massive, inpresence of a topological term or not -- retain their main properties when theThirring interaction is added or the gauge fields live in higher dimension.

String tension and robustness of confinement properties of in the Schwinger-Thirring model