Superconductivity from gauge/gravity duality with flavor

Abstract

We consider thermal strongly-coupled N=2 SYM theory with fundamental matter at finite isospin chemical potential. Using gauge/gravity duality, i.e. a probe of two flavor D7-branes embedded in the AdS black hole background, we find a critical temperature at which the system undergoes a second order phase transition. The critical exponent of this transition is one half and coincides with the result from mean field theory. In the thermodynamically favored phase, a flavor current acquires a vev and breaks an Abelian symmetry spontaneously. This new phase shows signatures known from superconductivity, such as an infinite dc conductivity and a gap in the frequency-dependent conductivity. The gravity setup allows for an explicit identification of the degrees of freedom in the dual field theory, as well as for a dual string picture of the condensation process.

Figures (5)

• Figure 1: Phase diagram for fundamental matter in thermal strongly-coupled ${\cal N}=2$ SYM theory Erdmenger:2008yj, with $\mu$ the isospin chemical potential, $M_q$ the bare quark mass, $\bar{M} = 2 M_q \lambda^{-1/2}$, $\lambda$ the 't Hooft coupling and $T$ the temperature: In the blue shaded region, mesons are stable. In the white and green regions, the mesons melt. Here the new phase is stabilized while it was unstable in Erdmenger:2008yj. In this phase we find some features known from superfluids.
• Figure 2: Dimensionless grand potential ${\cal W}_7$ (large plot) and specific heat ${\cal C}_7$ (small plot) versus temperature $T/T_c$. Blue: the unstable phase with $A^1_3\equiv 0$. Red: the new stable phase with $A^1_3\not\equiv 0$. Note that the dimensionful specific heat $C_7 \propto T^3 {\cal C}_7$ is rising above $T_c$.
• Figure 3: Sketch of our string setup: Strings spanned from the horizon of the AdS black hole to the D$7$-branes (green and blue plane) induce a charge at the horizon Erdmenger:2008yjKobayashi:2006sbKarch:2007br. In the setup considered here, there are also D$7$-D$7$ strings present as shown in the figure. These D$7$-D$7$ strings are distributed along the AdS radial coordinate $\varrho$, since they have to balance the flavorelectric and gravitational, i.e. tension forces. Thus these D$7$-D$7$ strings distribute the charges along the AdS radial coordinate, leading to a stable configuration of reduced energy. This corresponds to a superconducting condensate given by the Cooper pairs.
• Figure 4: Real part of the conductivity $\text{Re}\:\sigma$ versus frequency $\mathfrak{w}=\omega/(2\pi T)$ at different temperatures: $T\approx 0.90T_c$ (black), $T\approx 0.66T_c$ (green), $T\approx 0.46T_c$ (blue), $T\approx 0.28 T_c$ (red).
• Figure 5: Movement of quasinormal modes at increasing temperature $T$: The different colors indicate the different fluctuations ${\color{red}X}$, ${\color{green}Y}$ and ${\color{blue}a^3_2}$. A Higgs mechanism is evident as explained in the text.