Black hole in Dehnen $$\left( 1,4,\frac{1}{2}\right) $$ dark matter halo: exact solution, lensing, light ring, and thermodynamics

Abstract In this work, we present the exact solution of a static, spherically symmetric black hole embedded in a Dehnen $$\left( 1,4,\frac{1}{2}\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfenced> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>4</mml:mn> <mml:mo>,</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mfenced> </mml:math> dark matter halo. We explore both its optical and thermodynamic properties. Using the principle of least action, we derive the null geodesics to analyze and illustrate the corresponding gravitational lensing and light ring phenomena in the presence of the dark matter halo. We then turn to the thermodynamics of the black hole–dark matter system, examining the mass function, entropy, temperature, heat capacity, and Gibbs free energy to assess both local and global stability. Our results show that the surrounding dark matter halo plays a crucial role: it not only enhances the thermodynamic stability of the black hole but also enables the occurrence of phase transitions.