The Mermin-Wagner theorem states that continuous global symmetries cannot be broken in two or fewer spacetime dimensions; however, I have not seen this statement applied to gauge theories. Does it apply; ie, is there a Higgs mechanism for 1+1 QFTs?
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$\begingroup$ This question has already answered. Follow the link physics.stackexchange.com/questions/210488/…. And links therein $\endgroup$– MassJul 26, 2016 at 4:09
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1$\begingroup$ Gauge "symmetry" is not a symmetry but rather a redundancy in how you describe a system. Thus, there is no conserved current associated with it nor can you "break" it. This is discussed in detail here: physics.stackexchange.com/questions/13870/… $\endgroup$– AegonJul 26, 2016 at 21:46
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Gauge symmetry is actually not spontaneously broken in the Higgs mechanism; this is a common misconception. See What role does "spontaneously symmetry breaking" played in the "Higgs Mechanism"?. Therefore the Mermin-Wagner theorem does not apply to the Higgs mechanism, and the Higgs mechanism is possible in 1+1D.
