Effect of dark energy in geometrothermodynamics and phase transitions of regular Bardeen AdS black hole

We study the thermodynamics and geometrothermodynamics of regular Bardeen-AdS black hole with quintessence. The thermodynamics of the black hole is scrutinised using Temperature–Entropy (T–S), Pressure–Volume (P–v) and Gibbs energy plots, which indicates a critical behaviour. The behaviour is also confirmed from the divergence of specific heat against entropy, which shows a second-order phase transition. Furthermore, we observe that the quintessence state parameter ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega $$\end{document} shifts the transition point to lower entropy values. Using the concept of thermodynamic Ruppeiner and Weinhold geometry, we calculated the thermodynamic curvature scalar RR\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_R$$\end{document} and RW\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_W$$\end{document} in the quintessence dark energy regime (ω=-2/3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega = -2/3$$\end{document}). While these curvature scalars enable us to identify the critical behaviour, they do not show divergence at the phase transition points observed in specific heat study. To resolve this puzzle, we have adopted the method of geometrothermodynamics proposed by Hernando Quevedo. Choosing a Legendre invariant ‘Quevedo’ metric, the curvature scalar RQ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_Q$$\end{document} shows singularity at the same point as seen in the specific heat divergence.