Electric hyperscaling violating solutions in Einstein-Maxwell-dilaton gravity with $R^2$ corrections

In the context of holography applied to condensed matter physics, we study Einstein-Maxwell-dilaton theory with curvature squared corrections. This theory has three couplings ${\eta }_i$ for the three $R^2$ invariants and two theory functions: a dilaton potential $V(\varphi )$ and a dilaton-dependent gauge coupling $f(\varphi )$. We find hyperscaling violating (HSV) solutions of this theory, parametrized by dynamical critical exponent $z$ and HSV parameter $\theta$. We obtain restrictions on the form of the theory functions required to support HSV-type solutions using three physical inputs: the null energy condition, causality $z\ge 1$, and $d_{\text{eff}}\equiv d-\theta$ lying in the range $0<d_{\text{eff}}\le d$. The null energy condition constraints are linear in the ${\eta }_i$ and (quartic) polynomial in $d,z,\theta$. The allowed ranges of $z,\theta$ change depending on the signs of ${\eta }_i$. For the case of Einstein-Weyl gravity, we further narrow down the theory functions and solution parameters required for crossover solutions interpolating between HSV, ${\mathrm{AdS}}_{d+2}$ near the boundary, and ${\mathrm{AdS}}_2\times {\mathbb{R}}^d$ in the deep interior.

Figure 1

Restrictions on ${\eta }_{\mathrm{GB}}$ from the NEC for $d=2$ and $d=5$ respectively. Note that there is only one arrow toward decreasing ${\eta }_{\mathrm{GB}}$ for the $d=5$ case 1(b).

Figure 2

NEC restrictions on ${\eta }_W$ for $d=1$.

Figure 3

Restrictions on ${\eta }_W$ from the NEC for $d=2$. For $z<4$, the NEC is satisfied in the hexahedral region in the upper right-hand side of both subfigures. For $z>4$, the NEC is satisfied in the rectangular region in the lower left-hand side of both subfigures.

Figure 4

Restrictions on ${\eta }_R$ from the NEC for $d=2$ and $d=5$. Note that there are two black sides in both plots, unlike earlier cases.

Figure 5

Restrictions on ${\eta }_W$ and ${\eta }_R$ from the NEC for $d=2$ and several values of $z$. Notice the sharp change in behavior in Fig. 5 where $d=2$, $z=4$.

Figure 6

Restrictions on ${\eta }_{\mathrm{GB}}$ and ${\eta }_R$ from the NEC for $d=3$ and $z=2$ and $z=3$.

Figure 7

Restrictions on ${\eta }_{\mathrm{GB}}$ and ${\eta }_R$ from the NEC for $d=3$ and $z=4$ and $z=6$. Notice the sharp change in behavior compared (6) where $d=3$, $z<4$.

Figure 8

Restrictions on ${\eta }_{\mathrm{GB}}$ and ${\eta }_R$ from the NEC for $d=5$ and $z=2$ and $z=3$.

Figure 9

Restrictions on ${\eta }_{\mathrm{GB}}$ and ${\eta }_R$ from the NEC for $d=5$ and $z=4$ and $z=6$. Notice the sharp change in behavior compared to Fig. 8 where $d=5$, $z<4$.

Figure 10

Restrictions on ${\eta }_{\mathrm{GB}}$ and ${\eta }_W$ from the NEC for $d=3$ and $z=2$, $z=4$, $z=5$ and $z=6$.

Figure 11

Restrictions on ${\eta }_{\mathrm{GB}}$ and ${\eta }_W$ from the NEC for $d=4$ and $z=\frac{3}{2}$, $z=4$ and $z=6$.

Figure 12

Restrictions on ${\eta }_{\mathrm{GB}}$ and ${\eta }_W$ from the NEC for $d=5$ and $z=\frac{3}{2}$, $z=4$ and $z=6$.

Figure 13

Restrictions on ${\eta }_R$, ${\eta }_{\mathrm{GB}}$ and ${\eta }_W$ from the NEC for $d=3$ and $z=2$, and $\theta =1$ and $\theta =2$, respectively. Note the change in the allowed region when going from $\theta =1$ to $\theta =2$.

Figure 14

Restrictions on ${\eta }_R$, ${\eta }_{\mathrm{GB}}$ and ${\eta }_W$ from the NEC for $d=3$, $z=4$, $\theta =1$ and $d=3$, $z=6$, $\theta =2$, respectively.

Figure 15

Restrictions on ${\eta }_R$, ${\eta }_{\mathrm{GB}}$ and ${\eta }_W$ from the NEC for $d=5$ and $z=2$, and $\theta =1$, $\theta =3$, and $\theta =4$, respectively. Note the change in the allowed region when going from $\theta =1$ to $\theta =3$ and $\theta =4$.

Figure 16

Restrictions on ${\eta }_R$, ${\eta }_{\mathrm{GB}}$ and ${\eta }_W$ from the NEC for $d=5$, $z=4$, $\theta =4$ and $d=5$, $z=6$, $\theta =3$, respectively.