Thermodynamic topology of Einstein–Maxwell-dilaton theories

Abstract We present a systematic investigation of the thermodynamic topology for a broad class of asymptotically charged Anti-de Sitter (AdS) black holes in Einstein–Maxwell-Dilaton (EMD) theories, examining how scalar coupling parameters and spacetime dimensions influence black hole thermodynamics. Employing a topological approach that utilizes the torsion number of vector fields constructed from the generalized free energy, we characterize black hole states as topological defects within the thermodynamic parameter space. Through analytical solutions spanning dimensions $$d = 4,$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>=</mml:mo> <mml:mn>4</mml:mn> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> $$d=5,$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>=</mml:mo> <mml:mn>5</mml:mn> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> and $$d=6,$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>=</mml:mo> <mml:mn>6</mml:mn> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> including the Gubser–Rocha model, we demonstrate that variations in the dilaton coupling constant $$\delta ,$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>δ</mml:mi> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> particularly near its critical value $$\delta _c,$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>δ</mml:mi> <mml:mi>c</mml:mi> </mml:msub> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> induce transitions between distinct thermodynamic topological phases. Our analysis reveals that certain black hole solutions constitute a novel class designated as $$W^{0-\leftrightarrow 1+},$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>W</mml:mi> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>-</mml:mo> <mml:mo>↔</mml:mo> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> </mml:mrow> </mml:msup> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> characterized by a torsion number $$W = 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>W</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> that corresponds to a unique stability structure. We establish that Gubser–Rocha models belong to this topological classification. These results significantly expand the existing classification framework while reinforcing thermodynamic topology as a robust analytical tool for probing the universal properties of black holes in both gravitational and holographic contexts. The findings provide new insights into the relationship between microscopic couplings and macroscopic thermodynamic behavior in extended gravity theories.