Landscape of superconducting membranes

TL;DR

This paper establishes explicit string-theory realizations of holographic superconductivity in AdS$_4$/CFT$_3$ by identifying decoupled charged pseudoscalar modes arising from 3-form fluctuations in Sasaki–Einstein flux compactifications. It develops a general framework for holographic superconductivity, derives a concrete instability criterion $q^2 oldsymbol{eta}^2 \,\ge 3+2\boldsymbol{ riangle}(\boldsymbol{ riangle}-3)$, and computes Tc distributions across a landscape of ${ m N}=2$ M-theory vacua, including Brieskorn–Pham links and skew-whiffed backgrounds. The results show that many vacua exhibit superconducting instabilities at finite ${ m mu}$ with Tc ranging from sub-Kelvin to tens of Kelvins per millivolt, and they connect these instabilities to dual operators such as deformations of the superpotential. This work bridges the string landscape with condensed-matter quantum criticality, enabling a landscape-wide perspective on universal features of holographic superconductivity and suggesting pathways to identify the precise dual operators and endpoints in the gravity theory. All mathematical constructs are expressed within a holographic framework and highlight the potential to extend holographic superconductivity across AdS$_4$/CFT$_3$ (and beyond) within the broader string landscape.

Abstract

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 N=2 large N gauge theories in 2+1 dimensions, dual to Sasaki-Einstein compactifications of M theory, towards 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 one milliVolt, we find critical temperatures ranging between 0.24 and 165 degrees Kelvin.

Figures (3)

• 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 $\gamma T_c /\mu$. The BPS line $\Delta = \gamma q$ is shown in red; 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 ${{\Bbb Z}}_k$ quotients. The solutions have been binned into ranges of width $2$ K/mV.