Symmetry breaking in holographic theories with Lifshitz scaling

Abstract

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.

Are there Goldstone bosons in d≤z+1 ?

“…This is tantamount to say that the QFT under consideration, besides being in the large N limit, is also generically strongly coupled. We will use a set-up in all similar to the one considered in [16], though we will implement time-reversal symmetry to be consistent with the discussion in the previous section. On the bulk, gravity side of the holographic correspondence, we thus introduce a complex scalar φ charged under a Up1q gauge symmetry.…”

“…We have anticipated here the special case where d ď z`1 (i.e. 2z ěd), the opposite case was treated in [16]. Dots between leading and subleading orders mean that one can find some more terms by adding powers of r two by two, ifd´2 ą 2 and/or 2z´d ą 2.…”

“…We start by treating the other sectors. For the scalar sector, the procedure goes exactly as in [16]. We add the counterterm…”

“…We then explore what happens to spontaneous symmetry breaking in Lifshitz scaling theories which are described holographically (see [9][10][11][12][13][14][15][16]), i.e. in a large N limit.…”

“…in a large N limit. We extend the analysis of [16] to the above mentioned case of d ď z`1. As in the relativistic case [7], we find that alternative quantization for the vector is needed, albeit in the present case of Lifshitz scaling, only the temporal component of the vector is involved.…”