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I want to know what kinds of things theorists are currently looking at. Specifically, I want to know more about the promise that field-theoretic methods are showing. I am studying superconductivity at an advanced undergraduate/beginning graduate level. Links to review papers/papers appreciated. Thanks.

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First note that High $T_C$ superconductivity may not have a general mechanism that works for all material, in opposition to the case of regular superconductivity and BCS theory. I mention this because there is an historical interplay between superconductivity and magnetism in the matter, as in Meissner effect.

Currently there are (roughly) two families of high $T_C$ superconductors, the cuprate family and the iron pnictides, which are material with different magnetic properties. Therefore the most promising approach might depend on the material in question.

For the better studied case of the cuprates I would recommend this lectures from Tremblay. Not a review paper, so not really addressing what is on the edge, but very pedagogical introduction to the subject, noting what ideas seem relevant

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