Probing quantum criticality near the BTZ black hole horizon: Insights from coupled fermion-antifermion pairs

In this study, we analytically examine the behavior of a fermion-antifermion ( f f ‾ ) pair near the horizon of a static BTZ black hole using a fully covariant two-body Dirac equation with a position-dependent mass, m → m ( r ) . This formulation leads to a set of four first-order equations that can be reduced to a second-order wave equation, enabling the analysis of gravitational effects on quantum interactions. Two mass modifications are considered: (i) m → m − a / r , representing an attractive Coulomb interaction, and (ii) m → m − a / r + b r , corresponding to a Cornell potential. For case (i), an exact analytical solution is obtained, while for case (ii), conditionally exact solutions involving biconfluent Heun functions are derived. For the lowest mode ( n = 0 ), the results indicate that real oscillations without energy loss occur when a > 0.25 in scenario (i) and a > 0.75 in scenario (ii), suggesting stable oscillatory behavior. When a < 0.25 in scenario (i) or a < 0.75 in scenario (ii), the state exhibits decay, indicating instability below these critical thresholds. At a = 0.25 (scenario (i)) and a = 0.75 (scenario (ii)), the system reaches a state where its evolution ceases over time. These findings provide insights into the stability conditions of fermion-antifermion pairs near the black hole horizon and may have relevance for determining critical coupling strengths in systems such as holographic superconductors. Furthermore, this work adopts an effective semi-classical quantum gravity approach, offering a practical framework for incorporating gravitational effects. However, a more complete description of the system would require a deeper understanding of quantum gravity beyond computational methods. The results presented here may contribute to further studies exploring the influence of strong gravitational fields on quantum systems.