Flavor Superconductivity from Gauge/Gravity Duality

TL;DR

This work presents a top-down holographic model of flavor superconductivity driven by an isospin chemical potential in a strongly coupled N=2 theory, realized via two D7-branes in an AdS black hole background. A rho-meson condensate forms through a spontaneously broken U(1)_3, yielding a second-order phase transition with exponent 1/2 and a gap in the conductivity alongside infinite dc conductivity, characteristic of superconductivity. The authors provide two complementary methods to evaluate the non-Abelian DBI action—an adapted symmetrized trace and a fourth-order expansion—finding consistent thermodynamics and phase structure, and they illuminate the condensation mechanism with a string-theoretic picture of horizon-string recombination into D7-D7 strings. They also demonstrate fluctuations, transport properties, dynamical mass generation, and the Meissner effect in this flavor-superconducting phase, and discuss extensions to finite quark mass and backreaction. Overall, the study bridges gauge/gravity duality with explicit flavor dynamics to realize a rho-meson superfluid in a robust, high-energy framework with potential condensed matter analogies and future avenues for richer holographic superconductors.

Abstract

We give a detailed account and extensions of a holographic flavor superconductivity model which we have proposed recently. The model has an explicit field theory realization as strongly coupled N=2 Super Yang-Mills theory with fundamental matter at finite temperature and finite isospin chemical potential. Using gauge/gravity duality, i.e. a probe of two flavor D7-branes in the AdS black hole background, we show that the system undergoes a second order phase transition with critical exponent 1/2. The new ground state may be interpreted as a rho meson superfluid. It shows signatures known from superconductivity, such as an infinite dc conductivity and a gap in the frequency-dependent conductivity. We present a stringy picture of the condensation mechanism in terms of a recombination of strings. We give a detailed account of the evaluation of the non-Abelian Dirac-Born-Infeld action involved using two different methods. Finally we also consider the case of massive flavors and discuss the holographic Meissner-Ochsenfeld effect in our scenario.

Figures (13)

• 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 superconductivity.
• Figure 2: Sketch of our string setup: The figure shows the two coincident D7 branes stretched from the black hole horizon to the boundary as a green and a blue plane, respectively. Strings spanned from the horizon of the AdS black hole to the D$7$-branes induce a charge at the horizon Erdmenger:2008yjKobayashi:2006sbKarch:2007br. However, above a critical charge density, the strings charging the horizon recombine to D$7$-D$7$ strings. These D$7$-D$7$ strings are shown in the figure. Whereas the fundamental strings stretched between the horizon and the D$7$-brane are localized near the horizon, the D$7$-D$7$ strings propagate into the bulk balancing the flavorelectric and gravitational, i.e. tension forces (see text). Thus these D$7$-D$7$ strings distribute the isospin charges along the AdS radial coordinate, leading to a stable configuration of reduced energy. This configuration of D$7$-D$7$ strings corresponds to a superconducting condensate.
• Figure 3: Profiles of the relevant dimensionless gauge fields $\tilde{A}$ on the D$7$-branes and their dimensionless conjugate momenta $\tilde{p}$ versus the dimensionless AdS radial coordinate $\rho$ near the horizon at $\rho=1$. The different curves correspond to the temperature $T=T_c$ (blue) and $T\approx 0.9T_c$ (red). The plots are obtained at zero quark mass $m=0$ and by using the adapted symmetrized trace prescription. Similar plots may also be obtained at finite mass $m\not=0$ and by using the DBI action expanded to fourth order in $F$. These plots show the same features: (a) The gauge field $\tilde{A}_0^3$ increases monotonically towards the boundary. At the boundary, its value is given by the dimensionless chemical potential $\tilde{\mu}$. (b) The gauge field $\tilde{A}_3^1$ is zero for $T\ge T_c$. For $T<T_c$, its value is non-zero at the horizon and decreases monotonically towards the boundary where its value has to be zero. (c) The conjugate momentum $\tilde{p}_0^3$ of the gauge field $\tilde{A}_0^3$ is constant for $T\ge T_c$. For $T<T_c$, its value increases monotonically towards the boundary. Its boundary value is given by the dimensionless density $\tilde{d}_0^3$. (d) The conjugate momentum $\tilde{p}_3^1$ of the gauge field $\tilde{A}_3^1$ is zero for $T\ge T_c$. For $T<T_c$, its value increases monotonically towards the boundary. Its boundary value is given by the dimensionless density $-\tilde{d}_3^1$.
• Figure 4: The dimensionless grand canonical potential ${\cal W}_7$ calculated using the adapted symmetrized trace prescription versus temperature at zero quark mass $M_q=0$: Below $T=T_c$ the superconducting phase (red line) is thermodynamically preferred over the normal phase (blue line).
• Figure 5: Superconducting density $\tilde{d}_s=(\tilde{d}_0^3-c_0)/\tilde{d}_0^3$ versus temperature $T$: In both, the massless (red curve) and the massive case at $\mu/M_q=3$ (blue curve), the superconducting density $\tilde{d}_s$ vanishes linearly at the critical temperature. This is visualized by the fit $6.8 (1-T/T_c)$ (dashed blue curve).