Anisotropic gravity solutions in AdS/CMT

TL;DR

The paper shows how to realize Lifshitz-like fixed points in AdS/CMT by breaking Lorentz invariance to a Galilean-like subgroup, enabling anisotropic scaling across time and space with two exponents $z_1$ and $z_2$ and without requiring rotations. It constructs explicit bulk solutions in 2+1 and 3+1 dimensions using gravity coupled to form fields (including two-form and three-form fluxes) and analyzes the dual field theory via scalar operator dimensions, establishing a BF-type bound and the holographic two-point functions. The work demonstrates that Lifshitz points can arise from a minimal symmetry group (time/space translations plus scaling) and provides several consistent flux configurations that realize the desired anisotropies, broadening the holographic toolkit for non-relativistic quantum criticality. These results offer a pathway to model anisotropic quantum phases and potential connections to TL-liquid behavior within a holographic framework, with future directions aimed at deeper ties to condensed-matter systems and TL physics.

Abstract

We have constructed gravity solutions by breaking the Lorentzian symmetry to its subgroup, which means there is Galilean symmetry but without the rotational and boost invariance. This solution shows anisotropic behavior along both the temporal and spatial directions as well as among the spatial directions and more interestingly, it displays the precise scaling symmetry required for metric as well as the form fields. From the field theory point of view, it describes a theory which respects th5Ae scaling symmetry, $t\to λ^{z_1}t, x\to λ^{z_2}t, y\to λy$, for $z_1\neq z_2$, as well as the translational symmetry associated to both time and space directions, which means we have found a non-rotational but Lifshitz-like fixed points from the dual field theory point of view. We also discuss the minimum number of generators required to see the appearance of such Lifshitz points. In 1+1 dimensional field theory, it is 3 and for 2+1 dimensional field theory, the number is 4.