Below the Breitenlohner-Freedman bound in the nonrelativistic AdS/CFT correspondence

We propose that there is no analogue of the Breitenlohner-Freedman stability bound on the mass of a scalar field in the context of the nonrelativistic AdS/CFT correspondence. Our treatment is based on an equivalence between the field equation of a complex scalar in the AdS/CFT correspondence and the one-dimensional Schrödinger equation with an inverse square potential. We compute the two-point boundary correlation function for $m^2<m_{\mathrm{BF}}^2$ and discuss its relation to renormalization group limit cycles and the Efimov effect in quantum mechanics. The equivalence also helps to elucidate holographic renormalization group flows and calculations in the global coordinates for Schrödinger spacetime.

Figure 1

Graphical solution of Eq. (28) for $u_0={10}^{-4}$, $|\nu |=10$, and $M\omega =\frac{1}{4}$. The blue (red) line corresponds to the argument function of the left (right) hand side of Eq. (28). Points of intersection determine the discrete dimensionless eigenenergies ${ϵ}_l=\frac{{\overline{E}}_l}{4\omega M}$.

Figure 2

The phase portraits of the RG flows of the contact complex coupling $\lambda ={\lambda }_1+i{\lambda }_2$ in the one-dimensional inverse square potential problem: (a) undercritical regime $\kappa =\frac{1}{8}<{\kappa }_{\mathrm{cr}}$; (b) overcritical regime $\kappa =\frac{1}{2}>{\kappa }_{\mathrm{cr}}$. Arrows denote the direction towards the UV. In AdS/CFT case (a) corresponds to $m^2>m_{\mathrm{BF}}^2$, while (b) appears for $m^2<m_{\mathrm{BF}}^2$.