We consider a holographic setup where relativistic invariance is broken by a chemical potential, and a non-Abelian internal symmetry is broken spontaneously. We use the tool of holographic renormalization in order to infer what can be learned purely by analytic boundary considerations. We find that the expected Ward identities are correctly reproduced. In particular, we obtain the identity which implies the noncommutation of a pair of broken charges, which leads to the presence of Goldstone bosons with quadratic dispersion relations.
We investigate symmetry breaking in two-dimensional field theories which have
a holographic gravity dual. Being at large N, the Coleman theorem does not hold
and Goldstone bosons are expected. We consider the minimal setup to describe a
conserved current and a charged operator, and we perform holographic
renormalization in order to find the correct Ward identities describing
symmetry breaking. This involves some subtleties related to the different
boundary conditions that a vector can have in the three-dimensional bulk. We
establish which is the correct prescription that yields, after renormalization,
the same Ward identities as in higher dimensions.Comment: 20 pages. v2 comments added. Version to appear in JHE
Abstract:We study holographically Lifshitz-scaling theories with broken symmetries. In order to do this, we set up a bulk action with a complex scalar and a massless vector on a background which consists in a Lifshitz metric and a massive vector. We first study separately the complex scalar and the massless vector, finding a similar pattern in the twopoint functions that we can compute analytically. By coupling the probe complex scalar to the background massive vector we can construct probe actions that are more general than the usual Klein-Gordon action. Some of these actions have Galilean boost symmetry. Finally, in the presence of a symmetry breaking scalar profile in the bulk, we reproduce the expected Ward identities of a Lifshitz-scaling theory with a broken global continuous symmetry. In the spontaneous case, the latter imply the presence of a gapless mode, the Goldstone boson, which will have dispersion relations dictated by the Lifshitz scaling.
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