Cooper pairing near charged black holes

TL;DR

The study demonstrates that a quartic fermion interaction can induce Cooper pairing and a superconducting instability for charged fermions in a charged AdS black hole background, linking the critical temperature to a four-point fermion correlator in a strongly coupled 2+1D dual theory. By solving the massless Dirac equation and analyzing near-Fermi-surface dynamics, the authors derive an analytic expression for T_c that depends on the effective density of states N_eff and the bulk four-fermion coupling via T_c ∝ μ_RG e^{-M_F^2 L^2/N_eff}. The work shows Tc vanishes as the Fermi surface exponent ν approaches 1/2, and generalizes to multiple Fermi surfaces with a matrix eigenvalue problem for Tc. The dual-field-theory interpretation attributes the pairing strength to a specific fermionic four-point function, highlighting how holography encodes superconductivity emerging from strongly interacting non-Fermi liquids, while photon/graviton loop effects are suppressed in the large-N limit.

Abstract

We show that a quartic contact interaction between charged fermions can lead to Cooper pairing and a superconducting instability in the background of a charged asymptotically Anti-de Sitter black hole. For a massless fermion we obtain the zero mode analytically and compute the dependence of the critical temperature T_c on the charge of the fermion. The instability we find occurs at charges above a critical value, where the fermion dispersion relation near the Fermi surface is linear. The critical temperature goes to zero as the marginal Fermi liquid is approached, together with the density of states at the Fermi surface. Besides the charge, the critical temperature is controlled by a four point function of a fermionic operator in the dual strongly coupled field theory.

Figures (4)

• Figure 1: Fermi momentum vs charge of the fermion field. The dashed line is $\nu_k = 0$ and the shaded region is $0 < \nu_k < {{1\over2}}$. From left to right are the first, second, etc. Fermi surfaces, which disappear when they hit the dashed line. For the vertical axis, recall that $u_+ = \sqrt{3} \gamma/\mu$. Positive and negative $k_F$ correspond to Fermi surfaces in the Green's functions ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ respectively.
• Figure 2: Wavefunctions of the unstable mode $\Delta^{0}$ at different values of the charge for the first Fermi surface. From left to right $\gamma q = \{11, 5, 3, 2, 1.6 \}$. The curves are normalised so that their maxima are equal. At large charge the wavefunctions are supported away from the horizon. As $\nu \to {{1\over2}}$ at smaller charge ($\gamma q \to 1.56$), the wavefunctions are increasingly supported in the near horizon region.
• Figure 3: Left: Fermi velocity vs charge, equation (\ref{['vequation']}). $v_F$ vanishes at the dashed line when $\nu={{1\over2}}$. Right: $h_1$ vs charge, equation (\ref{['hequation']}). $h_1$ vanishes at the solid line when $\nu=0$. The multiple lines in each plot are for the various Fermi surfaces, in ascending order with the first Fermi surface on the left. Note that $v_F$ and $h_1$ have the same sign as $k_F$. As above, positive and negative $k_F$ correspond to Fermi surfaces in the Green's functions ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ respectively.
• Figure 4: Lighter lines show the effective density of states at the Fermi surface $N_\text{eff.}$ vs charge, for each Fermi surface with the first Fermi surface on the left. Recall from (\ref{['eq:Tc3']}) that $T_c \sim \mu e^{- M_F^2 L^2/N_\text{eff.}}$. The dark line is $N_\text{eff.}^\text{total}$, defined in (\ref{['multid']}), which accounts for the presence of multiple Fermi surfaces.