⚛️ high-energy theory

Complex Phase Structure and Widom line for Euler Heisenberg black holes

This paper investigates the supercritical thermodynamics of Euler-Heisenberg AdS black holes using Lee-Yang phase transition theory, revealing a complex phase structure with two distinct critical points and a degenerate higher-order critical point, and demonstrating how well-defined Widom lines emerge in the complex domain to act as effective stability boundaries even in the absence of conventional coexistence curves.

Original authors: Mozib Bin Awal, Prabwal Phukon

Published 2026-06-30

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Mozib Bin Awal, Prabwal Phukon

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a black hole not as a cosmic vacuum cleaner, but as a very strange, exotic fluid. Just like water can be ice, liquid, or steam, this black hole can exist in different "phases" depending on how hot it is and how much pressure is pushing on it.

This paper investigates what happens to these black holes when they get so hot and pressurized that they pass a specific tipping point, known as the critical point. In normal physics, once you pass this point, the distinction between liquid and gas disappears, and you get a "supercritical fluid." Scientists usually draw a line called the Widom line to mark where the fluid starts acting more like a liquid and where it starts acting more like a gas.

Here is the story of what the authors found, using simple analogies:

1. The Black Hole's "Personality" (The Euler-Heisenberg Model)

The black holes in this study are special because they interact with a specific type of electromagnetic force (nonlinear electrodynamics). Think of this force as a unique "personality trait" for the black hole. Depending on the strength of this trait, the black hole behaves in very different ways.

2. The Four-Phase Party

Most black holes have a simple transition: they switch from being "small" to "large." But these specific black holes are more complex. The authors found that under certain conditions, the black hole doesn't just have two phases; it has four distinct phases:

Smallest Black Hole
Small Black Hole
Intermediate Black Hole
Large Black Hole

Imagine a staircase with four distinct steps instead of just two. The black hole can stand on any of these steps, and it can jump between them.

3. The Two "Tipping Points"

Usually, a system has one critical point where the phases merge. However, this black hole system has two separate critical points.

Critical Point 1: Where the "Small" and "Large" phases merge.
Critical Point 2: Where the "Intermediate" and "Large" phases merge.

The authors discovered a special scenario where they could tune the black hole's parameters so that these two separate tipping points slide toward each other and crash into one another. This creates a Degenerate Critical Point. It's like two separate mountain peaks merging into a single, flat plateau.

4. The "Ghost" Map (Complex Phase Diagrams)

To understand what happens after these tipping points (the supercritical regime), the authors used a mathematical trick called Lee-Yang theory.

Think of the black hole's behavior as a map. On a normal map (real numbers), the lines stop at the critical point. But the authors decided to look at a "ghost map" (the complex plane).

In this ghost world, the lines don't just stop; they continue into a hidden dimension.
By looking at this hidden dimension, they could project a new line back onto our normal map. This projected line is the Widom line.

5. The Big Discovery: Widom Lines Without Coexistence

Here is the most surprising part of the paper:

The Normal Case: Usually, a Widom line is just the extension of a "coexistence curve" (the line where two phases, like liquid and gas, exist together). It's like a road that continues past a border.
The New Discovery: The authors found that for the Degenerate Critical Point and the Second Critical Point, there is no coexistence curve. There is no border where two phases sit together.
- Analogy: Imagine a road that suddenly ends. Usually, you can't draw a line past the end. But in this "ghost map," the authors found that a new road (the Widom line) appears out of nowhere, even though there was no border to extend from.

They found that this "ghost road" acts as a stability boundary. On one side of this line, the black hole is stable (it won't collapse or explode). On the other side, it is unstable. Even though there is no "phase transition" happening in the traditional sense, this invisible line still tells us where the black hole's behavior changes from safe to dangerous.

6. The Two Types of Widom Lines

In the regime where the black hole has two critical points, the authors found two different types of Widom lines coexisting:

The Traditional One: Connected to a phase transition (like the liquid-gas border).
The "Ghost" One: Born purely from the complex mathematical structure, with no underlying phase transition.

Summary

The paper shows that black holes are more complex than we thought. They can have multiple "tipping points" that merge together. Most importantly, it proves that the "Widom line" (a boundary marking a change in behavior) doesn't always need a traditional phase transition to exist. It can emerge purely from the deep, hidden mathematical structure of the system, acting as a safety line that separates stable black holes from unstable ones, even when no "phase change" is visibly happening.

Technical Summary: Complex Phase Structure and Widom Line for Euler-Heisenberg Black Holes

Problem Statement
While black hole thermodynamics has extensively explored phase transitions below the critical point (analogous to liquid-gas transitions), the supercritical regime remains less understood, particularly for systems with complex phase structures. Standard thermodynamic systems with a single critical point are typically characterized by a Widom line—a crossover region separating liquid-like and gas-like phases, often identified as the extension of a coexistence curve. However, systems with multiple critical points or degenerate criticalities may exhibit richer supercritical structures. The specific problem addressed in this work is the investigation of the supercritical thermodynamics of Euler-Heisenberg (EH) Anti-de Sitter (AdS) black holes, a system known to possess two distinct critical points and a degenerate higher-order critical point depending on the Euler-Heisenberg parameter $a$ . The authors aim to determine how the Lee-Yang theory of phase transitions, which encodes critical phenomena in the distribution of partition-function zeros in the complex plane, describes the phase structure and Widom lines in these regimes, specifically where conventional coexistence curves may be absent.

Methodology
The authors employ the Lee-Yang theory of phase transitions within the framework of extended phase space thermodynamics (black hole chemistry). The methodology involves:

Thermodynamic Setup: Starting from the Euler-Heisenberg action coupled to gravity, the authors derive the metric, Hawking temperature ( $T$ ), Gibbs free energy ( $G$ ), and specific heat ( $C_P$ ) for AdS black holes.
Critical Point Analysis: They solve the criticality conditions ( $\partial T/\partial r_+ = 0$ and $\partial^2 T/\partial r_+^2 = 0$ ) to identify distinct critical points. They further impose higher-order conditions ( $\partial^3 T/\partial r_+^3 = 0$ ) to locate a degenerate critical point where two ordinary critical points merge.
Complex Analytic Continuation: To probe the supercritical regime, the horizon radius $r_+$ is analytically continued into the complex plane. The Lee-Yang singularities are identified as the zeros of the partition function, which correspond to the divergences of the specific heat (poles of the second derivative of Gibbs free energy).
Phase Diagram Construction: The authors map the distribution of these complex singularities. The trajectories of these singularities in the complex $r_+$ plane are projected onto the real thermodynamic plane ( $P-T$ or $Q-T$ ) to define the Widom line. This line represents the real-plane manifestation of the underlying complex singularity structure.
Regime Comparison: The analysis is performed across four distinct regimes:
- Two-distinct-critical-point regime.
- Degenerate-critical-point regime (merger of the two).
- Single-critical-point regime.
- No-criticality regime.

Key Contributions and Results

Four-Phase Structure and Critical Points: The study confirms that for specific ranges of the Euler-Heisenberg parameter, the system exhibits a four-phase thermodynamic structure (Smallest, Small, Intermediate, and Large Black Holes). Solving the criticality equations yields two distinct sets of critical parameters. Furthermore, by tuning the pressure, these two critical points are shown to merge into a single degenerate critical point, representing a higher-order critical phenomenon.
Widom Lines in the Degenerate Regime: At the degenerate critical point, the system lacks a conventional first-order coexistence curve because only two branches (one stable, one unstable) exist. Despite this absence, the complex phase diagram reveals a well-defined Widom line. This line emerges from the complex singularity structure and acts as an effective stability boundary, separating thermodynamically stable ( $C_P > 0$ ) and unstable ( $C_P < 0$ ) sectors in the supercritical regime. The authors note that this Widom line is not the continuation of a coexistence curve but arises solely from the complex singularity structure.
Dual Widom Lines in the Two-Critical-Point Regime: In the regime with two distinct critical points, the complex phase diagram exhibits two distinct Widom lines:
1. Widom Line 1: Associated with the first critical point, this line is the supercritical continuation of a conventional coexistence curve, separating small and large black hole phases.
2. Widom Line 2: Associated with the second critical point, this line emerges despite the absence of a coexistence curve. It originates purely from the complex singularity structure and serves as a crossover boundary between distinct thermodynamic regimes.
  This finding demonstrates that Widom lines can exist independently of first-order phase transitions.
Consistency in Limiting Cases: The formalism is validated in limiting cases. In the single-critical-point regime, the analysis reproduces the standard picture of one coexistence curve and one Widom line. In the no-criticality regime (where parameters exceed the degenerate critical values), the complex singularities remain detached from the real axis, correctly indicating the absence of critical phenomena and phase transitions.

Significance and Claims
The paper claims that the Lee-Yang formalism provides a robust and consistent framework for exploring black hole thermodynamics in the supercritical regime, particularly for systems with intricate phase structures. The primary significance lies in the revelation that Widom lines can emerge without accompanying coexistence curves. The authors argue that the Widom line should not be viewed exclusively as the extension of a first-order phase boundary but rather as a manifestation of the underlying complex singularity structure of the Gibbs free energy.

The work suggests that the complex phase diagram offers a more complete description of the phase structure than real-valued diagrams alone, encoding crossover phenomena even when conventional thermodynamic singularities are absent. By identifying a degenerate critical point and the resulting unique Widom line structures, the study provides new insights into the origin and physical interpretation of Widom lines in gravitational thermodynamic systems, highlighting that supercritical crossovers need not possess a unique geometric origin tied to first-order transitions.