Fermionic quantum hair and topological phase transitions in regular black holes

Abstract This study investigates fermionic perturbations with vanishing total angular momentum (j=0) – corresponding to two coupled spinor fields – in regular black holes within asymptotically safe gravity. These spacetimes avoid singularities via regularization parameter $$\xi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ξ</mml:mi> </mml:math> . Using Rosen–Morse potential approximation and Chebyshev pseudo-spectral methods, we analyze quasi-normal modes (QNMs), thermodynamics, and topology. Results show increasing $$\xi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ξ</mml:mi> </mml:math> reduces the magnitude of imaginary QNM frequencies ( $$\omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ω</mml:mi> </mml:math> i), enhancing dynamical stability relative to singular black holes. The fermionic configuration introduces quantum hair through charge QD, modifying Hawking temperature ( $$\delta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>δ</mml:mi> </mml:math> TH $$\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>∞</mml:mi> </mml:math> $$\hbox {QD}^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mtext>QD</mml:mtext> <mml:mn>2</mml:mn> </mml:msup> </mml:math> / $$\hbox {rh}^3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mtext>rh</mml:mtext> <mml:mn>3</mml:mn> </mml:msup> </mml:math> ) and entropy ( $$\delta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>δ</mml:mi> </mml:math> S $$\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>∞</mml:mi> </mml:math> QD), potentially improving thermodynamic stability. Topologically, we identify a signature with a single phase transition characterized by winding number w = $$-1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> , yielding total topological charge W = $$-1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> – contrasting with W = 0 in bosonic analyses. Unlike previous studies of bosonic perturbations or fermions in singular spacetimes, this work bridges these domains by examining how specific fermionic configurations interact with nonsingular geometries. The interplay between $$\xi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ξ</mml:mi> </mml:math> , QNM structure, and thermodynamics reveals nuanced fermionic influences on black hole physics beyond classical relativity. These findings may inform observational searches for quantum gravity signatures through gravitational wave spectroscopy and accretion disk observations. Future research should extend to half-integer angular momentum cases (j = 1/2), corresponding to standard Dirac fields, explore massive spinor fields, and investigate fermion-boson interactions in regular black holes. This work establishes a framework for testing quantum-corrected black hole models while highlighting how particular fermionic degrees of freedom shape singularity-free gravitational systems. The parameter $$\xi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ξ</mml:mi> </mml:math> significantly impacts fermionic dynamics, suggesting quantum gravity effects could manifest through modified black hole stability and thermodynamics observable in astrophysical settings.