Noncommutative AdS black hole and the IR holographic superconductor

TL;DR

The paper develops a NC AdS$_4$-dyon black hole with planar horizon, where NC effects are encoded by a Gaussian-mass distribution via a Nicolini-like energy-momentum tensor. In a decoupling limit, the authors obtain analytic NC background solutions and a NC cutoff defined by the curvature, enabling controlled study of an infinitesimal scalar perturbation that models holographic superconductivity. They demonstrate that NC effects strengthen hair formation by sharpening the AdS$_2$ near-horizon throat, and they derive semi-analytic results for the upper critical magnetic field and BF bounds, including an explicit dependence on the NC nearness parameter $\alpha$. The analysis spans both canonical and grand canonical ensembles, revealing that NC corrections increase the condensate density and critical fields near the NC cutoff, while smoothly recovering the commutative results as $\alpha\to\infty$. Overall, the work provides a tractable, analytic handle on the IR (AdS$_2$) regime of NC holographic superconductors and highlights distinctive NC signatures in hair formation and magnetic response.

Abstract

We construct a noncommutative (NC) AdS$_4$-charged black hole with a planar horizon topology. The NC effects of this geometry are captured by a Gaussian distribution of black hole mass codified in a fluid-like energy-momentum tensor. A natural bound in radial coordinate is established, below which the scalar curvature changes its sign and defines a NC cutoff that embeds the point singularity. We study in detail the thermodynamic structure of this scenario, finding a well-defined black hole mass and an analytic criterion for its stability. Focusing on the AdS$_2$ structure near the horizon, we find a novel effective curvature radius with dependency on the NC cutoff. These results motivate us to explore the holographic superconducting system in terms of the nearness from the cutoff. The behavior of the magnetic field in the deep IR geometry is studied and we found semi-analytical novel expressions for the upper critical magnetic fields of a dual type-II superconductor in the canonical and grand canonical ensembles. The condensation in the form of hair is studied in terms of the bound states of the associated Schrödinger potential of the scalar field, interpreted as the dual to the density of Cooper pairs. The NC effects increase the hair formation due to a steeper AdS$_2$ throat comparable to the commutative case. Finally, we obtain the effective IR scalar field equation on the near-horizon and near-extremal NC Schwarzschild AdS$_2$ geometry and confirm that NC effects promote bound states that the commutative version forbids.

Figures (12)

• Figure 1: The blackening factor $\chi_{\hbox{\tiny NC}} (r)$ for different values of the noncommutative parameter $\theta$. It shows a continuous transition from the solution with two horizons, an extremal case, and no horizon structure. As $r\mapsto\infty$, the function $\chi_{\hbox{\tiny NC}}(r)\mapsto 1$, as it should be.
• Figure 2: Schematic picture of the global structure of the $(3+1)$-dimensional dyonic NC solution (\ref{['blackeningsol']}). The effective NC curvature in the near-horizon geometry acquires contributions of the nearness parameter $\alpha$. The NC effects on the curvature start at the value $r_{h}=2\sqrt{2\theta}$ ($\alpha=1$) where the effective curvature for C ($L^{\hbox{\tiny C}}_{\hbox{\tiny eff}}=L/\sqrt{6}$) and for NC, coincide. If we increase the $\alpha$ parameter slightly greater than one, $L^{\hbox{\tiny NC}}_{\hbox{\tiny eff}}$ decreases (red curve) until it converges again to the effective curvature for the commutative AdS$_{2}$. Far away from the horizon radius, the bare $L$ for AdS$_{4}$ dominates (i.e., $\alpha\gg1$); however, there are still reminiscences from the NC mass distribution.
• Figure 3: Noncommutative Euclidean renormalized on-shell action (taking $\mathcal{Q}=1$ and $r_{h}=1$) as a function of magnetic field $\mathcal{B}$. With $\alpha\gg1$, both actions coincide exactly, represented by the red curve. However, at $\alpha=1$, where the NC effects acquire more relevance, the on-shell action corresponds to the black curve.
• Figure 4: Determinant of the Hessian operator ($\vert H\vert$) as a function of $\alpha$ for different values of the electric charge. This function maintains its positivity above the value $\alpha\sim 1.05$. Note that this point acts as a fixed point, no matter how the parameters are varied.
• Figure 5: Fixing the horizon radius to unity, a comparison between extremal C (\ref{['bknC']}) and NC (\ref{['bknNC']}) blackening factors. The difference between the two functions seems to be minimal and, interestingly, for the blue curve, which corresponds to $\alpha\sim 1.25$, the difference between the functions is maximum, in contrast with the black curve for which $\alpha=1$. The convergence to the extremal C-function is very fast (red curve) at $\alpha=3$. The major difference occurs in the interval $r\in\left(1,1.8\right)$. Despite these minimal differences between C and NC functions, they have consequences in the holographic superconductor that we will explore in the next sections.