[Paper Review] Thermodynamic phase transition and winding number for the third-order Lovelock black hole

Yu-Shan Wang, Zhen-Ming Xu+1 more|arXiv (Cornell University)|Jul 4, 2023

Black Holes and Theoretical PhysicsPhysics and Astronomy48 references3 citations

TL;DR

This paper introduces a complex analysis method using the winding number to classify thermodynamic phase transitions in third-order Lovelock black holes, predicting coexisting first- and second-order transitions. It confirms predictions via thermal potential analysis, showing W=3 for hyperbolic and 7D spherical topologies (first- and second-order transitions), and W=4 for 7<d<12 spherical topologies, with distinct transition types depending on temperature regimes, validated by potential landscape behavior.

ABSTRACT

Phase transition is important for understanding the nature and evolution of the black hole thermodynamic system. In this study, the connection between the phase transition of a black hole and the winding number derived by the complex analysis is used to predict the type of the black hole phase transition. For the third-order Lovelock black holes, at the hyperbolic topology in any dimensions and the spherical topology in $7$ dimensions, we arrive at the winding numbers both are $W=3$ which predicts that the system will undergo both the first-order and second-order phase transitions. For the spherical topology in $7

Motivation & Objective

• To establish a topological invariant—winding number—from complex analysis to classify the order of thermodynamic phase transitions in third-order Lovelock black holes.
• To predict the coexistence of first- and second-order phase transitions in specific topological and dimensional regimes.
• To validate the winding number prediction using the thermal potential formalism, which models thermodynamic stability and transition pathways.
• To extend the understanding of black hole phase structure beyond standard swallowtail diagrams by incorporating topological invariants and potential landscapes.
• To provide a new framework linking complex geometry to black hole thermodynamics, offering deeper insight into microstructural evolution during phase transitions.

Proposed method

• The winding number W is computed from the complex mapping of the Hawking temperature T_h(r_h) over the entropy S(r_h), using the argument principle from complex analysis.
• The topological invariant W is interpreted as the number of Riemann surface foliations, with W=3 indicating a triple-fold structure and W=4 a quadruple-fold structure.
• The thermal potential U = ∫(T_h - T)dS is constructed to model thermodynamic stability, where extrema correspond to equilibrium states and curvature determines stability.
• The behavior of U(r_h) under varying pressure P and temperature T is analyzed to identify minima, maxima, and inflection points, signaling phase transitions.
• Critical temperatures T_cm and T_c2 are identified as transition thresholds between different phase transition regimes in spherical topologies for d>7.
• The correspondence between winding number predictions and thermal potential diagrams is used to validate the topological classification of phase transitions.

Experimental results

Research questions

• RQ1 How can the winding number from complex analysis be used to classify the order of thermodynamic phase transitions in third-order Lovelock black holes?
• RQ2 What is the topological structure (Riemann surface foliation) associated with different phase transition types in Lovelock black holes?
• RQ3 How does the thermal potential U(r_h) reflect the presence of first- and second-order phase transitions in varying dimensional and topological regimes?
• RQ4 Why does the phase transition behavior differ between k=-1 (hyperbolic) and k=+1 (spherical) topologies in higher dimensions?
• RQ5 What is the role of intermediate temperature T_cm in distinguishing between pure second-order and mixed first- and second-order transitions in d>7 spherical Lovelock black holes?

Key findings

• For hyperbolic topology in any dimension and spherical topology in 7 dimensions, the winding number is W=3, indicating the coexistence of first- and second-order phase transitions.
• In 7<d<12 dimensions with spherical topology, the winding number is W=4, corresponding to a four-foliating Riemann surface structure.
• For 7<d<12, when T_c1 < T < T_cm, the system exhibits only second-order phase transitions, corresponding to the decomposition 4=2+2.
• For 7<d<12, when T_cm < T < T_c2, the system undergoes both first- and second-order phase transitions, corresponding to the decomposition 4=1+3.
• Thermal potential analysis confirms that the U(r_h) plots exhibit two distinct minima at P=P_m, indicating a first-order transition, and show inflection points consistent with second-order transitions.
• The agreement between winding number predictions and thermal potential behavior validates the topological classification method as a reliable tool for analyzing complex black hole phase transitions.