Spatially modulated instabilities for scaling solutions at finite charge density

We consider finite charge density geometries which interpolate between ${\mathrm{AdS}}_2\times {\mathbb{R}}^2$ in the infrared and ${\mathrm{AdS}}_4$ in the ultraviolet, while traversing an intermediate regime of anisotropic Lifshitz scaling and hyperscaling violation. We work with Einstein-Maxwell-dilaton models and only turn on a background electric field. The spatially modulated instabilities of the near-horizon ${\mathrm{AdS}}_2$ part of the geometry are used to argue that the scaling solutions themselves should be thought of as being unstable—in the deep infrared—to spatially modulated phases. We identify instability windows for the scaling exponents $z$ and $\theta$, which are refined further by requiring the solutions to satisfy the null energy condition. This analysis reinforces the idea that, for large classes of models, spatially modulated phases describe the ground state of hyperscaling violating scaling geometries.

Figure 1

The dark (blue) shaded region represents the values of $z$ and $\theta$ compatible with NEC and associated with the spatially modulated instabilities predicted by (32) for case 1. The lighter (pink) shaded regions contains the remaining values of $\{z,\theta \}$ allowed by NEC but outside the instability window (32). We are plotting the range $-7<\theta <9$, $-3<z<7$.

Figure 2

The dark (blue) shaded region denotes the values of $z$ and $\theta$ compatible with NEC and susceptible to instabilities, for the special example described by (35). The light (pink) region contains the remaining values of $\{z,\theta \}$ allowed by NEC.