Critical phenomenon of quantum BTZ black holes

Abstract We extend the thermodynamics of quantum BTZ black holes by treating the quantum backreaction strength parameter $$\nu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ν</mml:mi> </mml:math> as a thermodynamic variable. We find various novel features. The critical point appears at $$\nu _c=1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>ν</mml:mi> <mml:mi>c</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> and a first order transition occurs either below the critical temperature for $$\nu &lt;\nu _c$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ν</mml:mi> <mml:mo>&lt;</mml:mo> <mml:msub> <mml:mi>ν</mml:mi> <mml:mi>c</mml:mi> </mml:msub> </mml:mrow> </mml:math> or above the critical temperature for $$\nu &gt;\nu _c$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ν</mml:mi> <mml:mo>&gt;</mml:mo> <mml:msub> <mml:mi>ν</mml:mi> <mml:mi>c</mml:mi> </mml:msub> </mml:mrow> </mml:math> . By solving the coexistence curve analytically, we analyze the phase structures and clarify an unexpected discontinuity around the critical point. The critical exponents are significantly different from the mean field theory results and violate one of the scaling laws. We present an intepretation for this by using a universal three scale factor hypothesis for critical behavior of thermodynamic potential. Finally, we prove that given an arbitrarily small angular momenta, only one stable black hole phase can exist and hence no transition will occur.