Constraining nonrelativistic RG flows with holography

We examine nonrelativistic holographic renormalization group (RG) flows by working with Einstein-Maxwell-scalar theories which support geometries that break Lorentz invariance at some energy scale. We adopt the superpotential formalism, which helps us characterize the radial flow in this setup and bring to light a number of generic features. In particular, we identify several quantities that behave monotonically under RG flow. As an example, we show that the index of refraction is generically monotonic. We also construct a combination of the superpotentials that flows monotonically in Einstein-scalar theories supporting nonrelativistic solutions and which reduces to the known c-function in the relativistic limit. Interestingly, such quantity also exhibits monotonicity in a variety of black hole solutions to the full Einstein-Maxwell-scalar theory, hinting at a deeper structure. Finally, we comment on the breakdown of such monotonicity conditions and on the relation to a candidate c-function obtained previously from entanglement entropy.

Figure 1

The superpotentials with respect to the radial coordinate $\overline{r}$ for the one-charge black hole at different temperatures. The inset shows the temperature as a function of ${\overline{r}}_h$ with ${\overline{r}}_h>1$. There are two branches of solutions for a given temperature above the minimum temperature $T_m=\sqrt{3/2}/\pi$ at ${\overline{r}}_h=\sqrt{5/3}$ (green). The superpotential $W$ (left panel) is monotonic in the branch with large ${\overline{r}}_h$, while nonmonotonic in the small ${\overline{r}}_h$ branch. In contrast, the quantity $W+\frac{1}{3}{W}_A$ (right panel) is always monotonic as a function of $\overline{r}$. We have worked in units with $\mu =1$.

Figure 2

The superpotentials with respect to the radial coordinate $\overline{r}$ for the two-charge black hole at different temperatures. The inset shows the temperature as a function of ${\overline{r}}_h$ with ${\overline{r}}_h>0$. There are two branches of solutions for a given temperature above the minimum temperature $T_m=\sqrt{3/2}/(2\pi )$ at ${\overline{r}}_h=\sqrt{1/6}$ (green). The superpotential $W$ (left panel) is monotonic in the branch with large ${\overline{r}}_h$, while nonmonotonic in the small ${\overline{r}}_h$ branch. In contrast, the quantity $W+\frac{1}{3}{W}_A$ (right panel) is always monotonic as a function of $\overline{r}$. We have worked in units with $\mu =1$.

Figure 3

The superpotentials with respect to the radial coordinate $\overline{r}$ for the three-charge black hole at different temperatures. The inset shows the temperature as a function of ${\overline{r}}_h$ with ${\overline{r}}_h\ge 0$ (the extremal case corresponds to the purple curve with ${\overline{r}}_h=0$). The extremal IR geometry is semi-local and is conformal to ${\mathrm{AdS}}_2\times R^2$. The superpotential $W$ (left panel) is not monotonic at low temperatures, but becomes monotonic when the temperature is increased. The combination $W+\frac{1}{3}{W}_A$ (right panel) is always monotonic as a function of $\overline{r}$. We have worked in units with $\mu =1$.

Figure 4

The superpotentials with respect to the radial coordinate $\overline{r}$ for the one-charge black hole in the STU model. The inset shows the temperature as a function of ${\overline{r}}_h$ with ${\overline{r}}_h>1$. There are two branches of solutions for a given temperature above the minimum temperature $T_m=\sqrt{2}/\pi$ at ${\overline{r}}_h=\sqrt{3/2}$ (green). The superpotential $W$ (left panel) is monotonic in the branch with large ${\overline{r}}_h$, while nonmonotonic in the small ${\overline{r}}_h$ branch. In contrast, the quantity $W+\frac{1}{4}{W}_A$ (right panel) is always monotonic as a function of $\overline{r}$. We have worked in units with $\mu =1$ and $L=1$.

Figure 5

The superpotentials with respect to the radial coordinate $\overline{r}$ for the two-charge black hole in the STU model at different temperatures. The inset shows the temperature as a function of ${\overline{r}}_h$ with ${\overline{r}}_h\ge 0$ (the extremal case corresponds to the purple curve with ${\overline{r}}_h=0$). The superpotential $W$ (left panel) is not monotonic at low temperatures, but it becomes monotonic when the temperature is increased. $W+\frac{1}{4}{W}_A$ (right panel) is always monotonic as a function of $\overline{r}$. We have worked in units with $\mu =1$ and $L=1$.

Figure 6

The superpotential $W+\frac{1}{3}{W}_A$ as a function of the radial coordinate $\overline{r}$ for the four-dimensional Reissner-Nordström black hole. It behaves nonmonotonically with respect to $\overline{r}$. We have worked in the units with $\mu =1$.

Figure 7

Numerical solutions corresponding to the parameter choice (a10). The solid blue curves denote the numerical solutions for the metric and scalar fields. The first plot shows that $X(u)$ and $Y(u)$ are identical, and thus the geometry is relativistic. The dot-dashed line in the fourth plot confirms that $X\sim u^2$ both in the UV and IR.

Figure 8

Numerical solutions corresponding to the parameters (a11). The solid blue curves correspond to the numerical solutions for the metric and scalar. The first plot confirms that the geometry is relativistic. There is a clean hyperscaling violating intermediate regime shown by the dashed red lines in the second and third plots. In the intermediate region the metric function $X$ scales as $\sim u^{50/41}$, while the scalar field has the expected logarithmic behavior. The two straight lines in the fourth plot confirm that $X\sim u^{50/41}$ in the intermediate region and $X\sim u^2$ both in the UV and IR. The scalar in the UV goes to zero as $\sim u^{-0.23}$, as we turn on a nontrivial source for the dual scalar operator with the scaling dimension $\mathrm{\Delta }\simeq 3.23$. In the last plot we have added the IR series solution (dotted orange curve) to check agreement with our numerical solution in the IR region.