Investigating quantum criticality through charged scalar fields near the BTZ black hole horizon

We examine a charged scalar field with a position-dependent mass $m\left(\rho \right)=m_0+S\left(\rho \right)$, where $S\left(\rho \right)$ represents a Lorentz scalar potential, near a BTZ black hole in the presence of an external magnetic field. By deriving the Klein–Gordon equation for this setup, we explore two scenarios: (i) a mass-modified scalar field with $m\left(\rho \right)=m_0-a/\rho$ (an exactly solvable case), and (ii) a scenario involving both mass modification and an external magnetic field (conditionally exactly solvable). We identify quantum critical points (QCPs) associated with the coupling constant $a$. In the first scenario, for massless charged scalar fields, critical points occur at $a=n+1/2$ for all radial quantum numbers $n\ge 0$ and magnetic quantum numbers $\left|m\right|\ge 0$. In the second scenario, these critical points shift to $a=n+3/2$ for $n\ge 0$ and $\left|m\right|>0$, with the case $m=0$ excluded. For massive scalar fields, QCPs emerge at $a=(n+1/2)/2$, leading to non-propagating fields at zero frequency. At these QCPs, the field frequencies drop to zero, marking transitions from stable oscillatory modes to non-propagating states. Below the critical points, the system exhibits instability, characterized by negative imaginary frequencies that suggest rapid decay and high dissipation. Above the critical points, the modes stabilize and propagate, indicating a transition to a superconducting-like phase, where dissipation vanishes and stable excitations dominate.