Landscape of superconducting membranes

The AdS/CFT correspondence may connect the landscape of string vacua and the “atomic landscape” of condensed matter physics. We study the stability of a landscape of IR fixed points of $\mathcal{N}=2$ large $N$ gauge theories in $2+1$ dimensions, dual to Sasaki-Einstein compactifications of M theory, toward a superconducting state. By exhibiting instabilities of charged black holes in these compactifications, we show that many of these theories have charged operators that condense when the theory is placed at a finite chemical potential. We compute a statistical distribution of critical superconducting temperatures for a subset of these theories. With a chemical potential of 1 mV, we find critical temperatures ranging between 0.24 and 165 K.

Figure 1

The critical temperature $T_c$ for a minimally coupled scalar as a function of the charge $\gamma q$ and dimension $\Delta$ of the dual operator. Contours are labeled by values of $\gamma T_c/\mu$. This plot is obtained numerically from Eq. (11). The BPS line $\Delta =\gamma q$ is shown as a heavy black (red) line; the shaded triangle to the left of it is the window of unstable values compatible with the BPS bound. The top boundary $q^2{\gamma }^2=3+2\Delta (\Delta -3)$ is a line of quantum critical points separating superconducting and normal phases at $T=0$. The bottom boundary is the unitarity bound $\Delta =1/2$, where $T_c$ diverges. The black dots indicate special cases which we will see arise in the context of $\mathcal{N}=2$ M2 brane theories.

Figure 2

Critical temperature $\gamma T_c/\mu$ as a function of $\Delta$ for operators on the BPS line $\Delta =\gamma q$.

Figure 3

A logarithmic distribution of critical temperatures over the chemical potential, in units of degrees Kelvin per milliVolt. The distribution is obtained from a scan over Brieskorn-Pham cones admitting Sasaki-Einstein metrics with moduli, along with allowed ${\mathbb{Z}}_k$ quotients. The solutions have been binned into ranges of width 2 K/mV.