Symmetry breaking in holographic theories with Lifshitz scaling

TL;DR

This paper investigates how symmetries are broken in Lifshitz-scaling holographic theories by analyzing bulk actions with scalar and vector fields, revealing the emergence of Goldstone modes with Lifshitz dispersion.

Contribution

It introduces a new holographic setup with complex scalar and vector fields on Lifshitz backgrounds, demonstrating symmetry breaking and Goldstone bosons with Lifshitz scaling.

Findings

Analytical two-point functions for scalar and vector fields.

Construction of probe actions with Galilean boost symmetry.

Reproduction of Ward identities and Goldstone modes in Lifshitz theories.

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 two-point 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.