# Holographic superconductor [[0001 AdS-CFT|Holographically]], by studying the bulk saddles as the boundary parameters are varied, the phase structure in strongly coupled field theories can be mapped out. Holographic superconductors are found this way, where -- below a critical temperature -- a phase transition of the bulk saddle happens, and it corresponds to a superconducting phase transition in the boundary system. The basic idea is similar to that of a [[0012 Hawking-Page transition|Hawking-Page transition]], but the boundary variables here are richer due to the bulk theory containing more fields and therefore more asymptotic data. A minimal model for holographic superconductivity has the following Lagrangian (in AdS${}_4$): $\mathcal{L}=R+\frac{6}{L^2}-\frac{1}{4} F^{a b} F_{a b}-V(|\psi|)-|\nabla \psi-i q A \psi|^2,$where $V(\psi)=-\frac{2}{L^2}\psi^2$. Below a critical temperature, $T_c$, the bulk saddle is a hairy black hole, i.e., one where the mater field (charged scalar) has a non-vanishing profile. This is the superconducting phase. Above this temperature, the solution is the usual RN black hole with a vanishing scalar profile. ## Refs - originals - [[Gubser2008]][](http://arxiv.org/abs/arXiv:0801.2977) - gives an argument - [[HartnollHerzogHorowitz200803]] - large charge limit -> no backreaction - [[2008#Hartnoll, Herzog, Horowitz (Oct)]] - arbitrary charge and backreaction considered - reviews - [[Hartnoll2009Lectures]][](https://arxiv.org/abs/0903.3246) - [[Herzog2009Lectures]][](https://arxiv.org/abs/0904.1975) - [[Horowitz2010Lectures]][](https://arxiv.org/abs/1002.1722) - [[Hartnoll2011BookChapter]] - extensions - rotations: [[Sonner2009]][](https://arxiv.org/abs/0903.0627) - zero-temperature limit: - [[2009#Horowitz, Roberts]] - [[KonoplyaZhidenko2009]][](https://arxiv.org/abs/0909.2138) - [[0006 Higher-derivative gravity|higher derivative corrections]]: - [[0425 Gauss-Bonnet gravity|GB gravity]] [[GregoryKannoSoda2009]][](https://arxiv.org/abs/0907.3203) [[PanWangPapantonopoulosOliveiraPavan2009]][](https://arxiv.org/abs/0912.2475) [[CaiNieZhang2010]][](https://arxiv.org/abs/1007.3321) - Weyl - [[WuCaoKuangLi2010]][](https://arxiv.org/abs/1010.1929) - [[MomeniSetare2011]][](https://arxiv.org/abs/1106.0431) - [[MomeniMajdMyrzakulov2012]][](https://arxiv.org/abs/1204.1246) - Josephson junction: [[HorowitzSantosWay2011]][](https://arxiv.org/abs/1101.3326) - Kondo effect: [[ErdmengerHoyosOBannonWu2013]][](https://arxiv.org/abs/1310.3271) - Meissner effect: [[NatsuumeOkamura2022]][](https://arxiv.org/pdf/2207.07182.pdf) - embedding in string theory - [[GauntlettSonnerWiseman200907]][](https://arxiv.org/abs/0907.3796) and [[GauntlettSonnerWiseman200912]][](https://arxiv.org/abs/0912.0512) - [[GubserHerzogPufuTesileanu2009]][](http://arxiv.org/abs/0907.3510) ## Related - [[0455 Black hole uniqueness theorems]] - [[0495 Hairy black holes]]