Resolving Lifshitz Horizons

TL;DR

This work shows that in simple Einstein-Maxwell-dilaton models, quantum corrections to the gauge coupling function $f(\phi)$ generate an infrared $AdS_2 \times \mathbb{R}^2$ attractor, replacing the problematic Lifshitz horizon and curing the mild singularity. The authors derive exact $AdS_2 \times \mathbb{R}^2$ solutions with a stabilized dilaton $e^{-\alpha \phi_H} = \xi_2^{-1/4}$ and compute the IR scale and magnetic charge, then construct RG flows from an $AdS_4$ UV through a wide Lifshitz regime to the IR fixed point. Numerical analyses show these flows exist for broad parameter choices, yielding a three-region structure: UV $AdS_4$, intermediate Lifshitz-like scaling, and IR $AdS_2 \times \mathbb{R}^2$. The results indicate a generic mechanism by which Lifshitz horizons are resolved by quantum effects, with implications for ground-state structure and phase diagrams in holographic condensed matter setups, and motivate further exploration in string-theory embeddings.

Abstract

Via the AdS/CFT correspondence, ground states of field theories at finite charge density are mapped to extremal black brane solutions. Studies of simple gravity + matter systems in this context have uncovered wide new classes of extremal geometries. The Lifshitz metrics characterising field theories with non-trivial dynamical critical exponent $z \neq 1$ emerge as one common endpoint in doped holographic toy models. However, the Lifshitz horizon exhibits mildly singular behaviour - while curvature invariants are finite, there are diverging tidal forces. Here we show that in some of the simplest contexts where Lifshitz metrics emerge, Einstein-Maxwell-dilaton theories, generic corrections lead to a replacement of the Lifshitz metric, in the deep infrared, by a re-emergent $AdS_2 \times R^2$ geometry. Thus, at least in these cases, the Lifshitz scaling characterises the physics over a wide range of energy scales, but the mild singularity is cured by quantum or stringy effects.

Figures (3)

• Figure 1: Metric functions in the intermediate Lifshitz regime. The numerical solution is plotted in blue while the behavior of an exact Lifshitz solution (with $\alpha=1$) is shown in red. On the left is $a'(r)$, and on the right is $b(r)$.
• Figure 2: Here we see a log-log plot of the crossover from Lifshitz scaling to $AdS_4$ in the metric functions $a(r)$ and $b(r)$. The crossover occurs around $r=10^{11}$. For $r<10^{11}$, the Lifshitz region persists over several decades in $r$, while for $r>10^{11}$, the solution becomes $AdS_4$. Left: $a'(r)$; right: $b(r)$. The flow in $a(r)$ just reflects the fact that the coefficient of the linear term in $a(r) \sim r$ is different in the Lifshitz and $AdS_4$ regions. The change in slope in the log-log plot for $b(r)$ indicates the difference between a solution with dynamical scaling ($z = 5$, for our choice of parameters) and the $z=1$ characteristic of $AdS_4$.
• Figure 3: Left: Shown in blue is the numerical solution for $\phi(r)$ in the Lifshitz scaling regime, with the exact Lifshitz solution shown in red. Right: Log-log plot of $\phi(r)$ showing crossover from Lifshitz scaling to $AdS_4$. As in Figure (\ref{['fig:abcrossover']}), the crossover occurs around $r=10^{11}$, where we see $\phi(r)$ transition from a log-running function to a constant solution $\phi_\infty$.