Gauge gravity duality for d-wave superconductors: prospects and challenges

Abstract

We write down an action for a charged, massive spin two field in a fixed Einstein background. Despite some technical problems, we argue that in an effective field theory framework and in the context of the AdS/CFT correspondence, this action can be used to study the properties of a superfluid phase transition with a d-wave order parameter in a dual strongly interacting field theory. We investigate the phase diagram and the charge conductivity of the superfluid phase. We also explain how possible couplings between the spin two field and bulk fermions affect the fermion spectral function.

Figures (6)

• Figure 1: (Color online) The order parameter $\langle \mathcal{O}_{xy} \rangle$ in $d=3$ spacetime dimensions as a function of the temperature for various values of $\Delta$ which decrease from top to bottom. The chemical potential $\mu$ and the charge density $\rho$ were obtained numerically using standard methods. The units we use to measure $\langle \mathcal{O}_{xy} \rangle$ are similar to those used in Yarom:2009uq.
• Figure 2: (Color online) The real part of the conductivity as a function of frequency for a $d$-wave condensate of conformal dimension $\Delta=4$. As the temperature is decreased one observes a spike in the conductivity indicative of a bound state. This spike is localized at smaller values of $\omega$ as the temperature is lowered. A second spike in the conductivity appears to vanish as the temperature is decreased.
• Figure 3: (Color online) The fermion spectral function for decoupled ($\eta = 0$) massless fermions at $T=0.66 T_c$. Red implies a large value of the spectral function and blue represents a value closer to zero. The red bands in the spectral function are associated with quasi-normal modes of $\zeta_1$ and $\zeta_2$. As described in the text, if the dispersion relation of $\zeta_1$ is $\omega = E(k_x)$ then that of $\zeta_2$ is $\omega = E(-k_x)$.
• Figure 4: (Color online) The spectral function for the fermions evaluated for $\eta=0.5$ and $T=0.66 T_c$. The red bands corresponds to a large value of the spectral function and the blue regions to values which are closer to zero. The left panel corresponds to the anti-node, the central panel to a $22.5$ degree angle from the node and the value of the spectral function at the node is shown in the right panel.
• Figure 5: The Feynman diagrams associated with the interaction terms between the fermions and the spin two field in (\ref{['E:Fterms']}): (a) $\eta^* \varphi^*_{\mu\nu} \overline{\Psi^c}\Gamma^{\mu}D^{\nu} \Psi$; (b) $\eta \overline{\Psi} \Gamma^{\mu} D^{\nu} \left(\varphi_{\mu\nu}\Psi^c\right)$.