The strong CP problem arises from the Standard Model of particle physics, our current best description of fundamental forces and particles. The theory allows for a term in the quantum chromodynamics (QCD) sector that violates CP symmetry, meaning it distinguishes between matter and antimatter. However, experimental observations show that this CP violation is extraordinarily small, almost vanishingly so. This stark discrepancy between theoretical prediction and experimental reality is the essence of the strong CP problem. Without a mechanism to suppress this term, the vacuum of the universe would be permeated with a phenomenon called an electric dipole moment for the neutron, a property that has been meticulously searched for and not found to any significant degree. This profound absence of observable CP violation in the strong interactions suggests that our current understanding is incomplete, pointing towards new physics beyond the Standard Model that actively enforces this near-perfect symmetry.

Enter the Peccei-Quinn mechanism, a celebrated theoretical solution proposed in the late 1970s. This elegant idea introduces a new global symmetry, broken at a very high energy scale, which dynamically generates a very small coefficient for the problematic CP-violating term in QCD. The breaking of this Peccei-Quinn symmetry is accompanied by the emergence of a new, very light, and very weakly interacting particle known as the axion. The axion, in this context, is not merely a byproduct of solving the strong CP problem; it is an integral part of the solution. Its extremely weak interactions and low mass make it an ideal candidate for making up a significant fraction of the dark matter in the universe, thus connecting the solution to one fundamental problem with the potential to unravel another equally perplexing cosmic mystery.

The challenge, however, has always been to embed the Peccei-Quinn mechanism within a consistent and realistic theoretical framework that respects the symmetries observed in nature while also explaining the immense energy scale at which this symmetry breaking must occur to naturally suppress the neutron electric dipole moment to observed levels. Previous attempts often relied on specific particle content and symmetry structures that faced their own theoretical or experimental hurdles. The beauty of the new research lies in its ingenious approach: it proposes to generate a high-quality Peccei-Quinn symmetry not through a single, monolithic symmetry breaking, but through a sophisticated interplay of vertical and horizontal gauge symmetries. This distinction is crucial to understanding the novelty of the proposed solution.

Gauge symmetries are a cornerstone of modern physics, dictating the fundamental forces. In the context of grand unified theories (GUTs), which aim to unify the electroweak and strong forces at high energies, these symmetries are often described in terms of how they act on different generations of particles and how they are broken down to the symmetries of the Standard Model. Vertical symmetries typically relate to how particles transform within multiplets of a given gauge group. Horizontal symmetries, on the other hand, often relate to symmetries that act between different generations of fermions or relate particles with different quantum numbers in a way that preserves the vertical structure. The interplay described in the paper suggests a dynamic where the breaking of these distinct types of symmetries conspires to naturally provide the necessary conditions for the Peccei-Quinn symmetry to be well-behaved.

The precise details of this interplay are highly technical, involving concepts like discrete symmetries, flavor symmetries, and radiative symmetry breaking. The researchers have constructed a model where the spontaneous breaking of a large gauge group, which encompasses both vertical and horizontal symmetries, leads to the emergence of distinct symmetry breaking scales. It is this layered approach to symmetry breaking that appears to be the key. Instead of a single, enormous energy scale for Peccei-Quinn symmetry breaking, which can sometimes lead to fine-tuning problems and other theoretical difficulties, this model proposes a situation where the effective breaking scale required for the axion to solve the strong CP problem is naturally generated from the combined effects of these different gauge symmetry breakings.

Imagine a complex machine with multiple interlocking gears. The overall motion of the machine is not determined by any single gear, but by the precise interaction and relative speeds of all of them. Similarly, in this theoretical model, the high-quality Peccei-Quinn symmetry and the consequent suppression of CP violation are not the result of a single grand event but a carefully orchestrated consequence of the breaking of higher-dimensional gauge symmetries that govern the interactions and transformations of fundamental particles at very high energies. This cascading effect of symmetry breaking is what allows for the Peccei-Quinn symmetry to be “high-quality,” meaning it effectively suppresses the unwanted CP violation without requiring ad hoc adjustments.

The implications of this work are far-reaching. If this theoretical framework correctly describes the underlying physics, it not only offers a compelling resolution to the strong CP problem but also strongly suggests the existence of axions. As mentioned, axions are compelling dark matter candidates. Their mass and interaction strength can be tuned by the energy scale of Peccei-Quinn symmetry breaking. A high-quality PQ symmetry, as proposed, would imply axions with properties that align with cosmological observations of dark matter. This would be a monumental achievement, linking the solution to one of particle physics’ most persistent puzzles with the solution to one of cosmology’s most significant mysteries in a unified theoretical framework.

The concept of “high-quality” Peccei-Quinn symmetry is critical here. In some models, the PQ symmetry might be too weak or break at too low an energy scale, failing to adequately suppress the neutron electric dipole moment. Alternatively, it might break at such an enormous scale that it becomes difficult to reconcile with other aspects of particle physics. The proposed mechanism, by leveraging the interplay of vertical and horizontal gauge symmetries, is claimed to naturally generate an effective PQ breaking scale that is neither too high nor too low, leading to the precisely desired level of CP symmetry violation suppression. This naturalness is a highly coveted feature in theoretical physics, as it avoids the need for artificial fine-tuning of parameters.

Furthermore, the model’s reliance on gauge symmetries is significant. Gauge symmetries are fundamental to our understanding of fundamental forces. Theories that are built upon well-motivated gauge structures, especially those that aim for unification of forces, are often considered more robust. The inclusion of both vertical and horizontal gauge symmetry breaking suggests a richer and more complex underlying structure than previously considered, which could have implications for other areas of particle physics, such as fermion mass hierarchies and mixing patterns, which are themselves areas of active research and ongoing puzzles. This model could potentially offer a unified perspective on several disparate problems.

The energy scales involved in the breaking of these symmetries are expected to be extraordinarily high, far beyond the reach of current particle accelerators like the Large Hadron Collider. This means that direct experimental verification of the proposed gauge symmetry structure will be challenging. However, the predicted existence of axions opens up a new frontier for experimental searches. These axions, if they constitute dark matter, would interact exceedingly weakly with ordinary matter, but their unique properties could be detectable through specialized experiments designed to look for their characteristic signatures, such as resonant conversion into photons in strong magnetic fields. The precision of these future experiments may finally be able to probe the very low-mass, weakly interacting particles predicted by axion models.

The research also highlights the power of theoretical model building in extending our understanding of fundamental physics. Faced with experimental hints of new physics – the absence of nucleon electric dipole moments and the existence of dark matter – theorists are compelled to construct new frameworks. This paper exemplifies how exploring complex symmetry structures can lead to elegant solutions. The intricate dance of vertical and horizontal gauge symmetries, a concept that might seem abstract, is demonstrated to have profound consequences for the fundamental properties of our universe, from the behavior of subatomic particles to the composition of the cosmos itself. This exemplifies how seemingly esoteric mathematical constructs can have tangible physical implications.

The implications for dark matter research are particularly exciting. If axions are indeed the dark matter, then understanding the Peccei-Quinn mechanism and its origin becomes paramount to understanding the nature of dark matter. This research provides a compelling new avenue for generating these axions. It suggests that the dark matter we observe might not be some exotic, entirely new type of particle, but rather a natural consequence of a mechanism that elegantly solves another long-standing puzzle in fundamental physics. This is the kind of theoretical synergy that drives scientific progress, elegantly tying together seemingly unrelated phenomena into a coherent picture, a testament to the interconnectedness of the fundamental laws governing reality.

The mathematical rigor and the detailed construction of the theoretical model are crucial. The paper meticulously outlines the group theory, the symmetry breaking patterns, and the calculations that lead to the desired outcome. This level of detail is what allows the scientific community to scrutinize the proposal, identify potential weaknesses, and explore alternative avenues. The scientific process thrives on such rigorous proposals, which serve as springboards for further investigation, experimental design, and refinement of theoretical understanding. The strength of this work lies in its detailed and verifiable theoretical framework, which invites further study and critical analysis from the global physics community.

Ultimately, this research represents a significant step forward in our quest to understand the fundamental constituents and forces of the universe. By proposing a novel way to generate a high-quality Peccei-Quinn symmetry through the interplay of vertical and horizontal gauge symmetries, the authors have opened a new window into the possible origins of the universe’s near-perfect CP symmetry and, quite possibly, the nature of dark matter. It is a testament to the enduring power of theoretical physics to tackle the most profound mysteries, pushing the boundaries of our knowledge and guiding the path for future experimental exploration. The quest to unify our understanding of the cosmos continues, driven by such elegant and insightful theoretical advancements.

This research delves into the heart of fundamental physics, offering a sophisticated solution to the notorious strong CP problem that has puzzled physicists for decades. The Standard Model, while incredibly successful, contains a theoretical term in its description of the strong force that predicts a non-zero electric dipole moment for the neutron. However, experimental searches have consistently found this value to be incredibly small, almost zero. This glaring discrepancy, the strong CP problem, suggests that our current understanding is incomplete. The Peccei-Quinn mechanism was proposed to elegantly address this issue by introducing a new symmetry that, when broken, naturally suppresses this problematic CP-violating term.

The key innovation of the Di Luzio et al. paper lies in how they propose this Peccei-Quinn symmetry is established. Instead of relying on a single, high-energy symmetry breaking event, they introduce a framework based on the complex interplay of “vertical” and “horizontal” gauge symmetries. These terms refer to different ways in which fundamental particles and forces can be related and transformed at extremely high energies, far beyond what current accelerators can probe. The intricate interaction and subsequent breaking of these distinct gauge symmetries, as described in their model, are precisely orchestrated to generate an effective Peccei-Quinn symmetry that is “high-quality” – meaning it effectively solves the strong CP problem without requiring unnatural fine-tuning of parameters.

This proposed mechanism is particularly exciting because it offers a strong theoretical motivation for the existence of axions. When the Peccei-Quinn symmetry is broken, it predicts the emergence of a very light and very weakly interacting particle called an axion. For decades, axions have been a leading candidate for dark matter, the invisible substance that makes up the vast majority of matter in the universe but remains elusive to direct detection. If the axion is indeed the dark matter, then understanding the specific properties of the Peccei-Quinn symmetry and its breaking mechanism is crucial for understanding the nature of dark matter itself. This research offers a robust theoretical pathway for generating axions with properties consistent with cosmological observations of dark matter.

The technical details involve a sophisticated understanding of gauge theories, grand unification concepts, and spontaneous symmetry breaking. The researchers have carefully constructed a model where a specific arrangement of gauge groups and their breaking down to the Standard Model symmetries naturally leads to the formation of a stable vacuum state that respects approximate CP symmetry in the strong interactions. The concept of “vertical” symmetries might relate to how particles within a generation transform under a gauge group, while “horizontal” symmetries could relate transformations between different generations or particle types in a manner that is crucial for generating the desired Peccei-Quinn structure. This duality in symmetry breaking is the lynchpin of their argument.

The absence of a detectable neutron electric dipole moment has been a significant puzzle. The theoretical Peccei-Quinn mechanism provides a compelling explanation, but its implementation within a realistic model has always been a challenge. This new work elegantly sidesteps some of the difficulties encountered in previous models. By proposing a composite mechanism for generating the Peccei-Quinn symmetry from the interplay of distinct gauge symmetries, they achieve a scenario where the symmetry is naturally well-behaved, leading to the correct suppression of CP violation without resorting to unnatural fine-tuning of fundamental constants. This quest for “naturalness” is a driving force in theoretical physics.

The scientific community will undoubtedly scrutinize this model with great interest. The proposed mechanism, while theoretically sound, relies on physics at energy scales far beyond our current experimental reach. However, the prediction of axions as dark matter candidates provides a clear target for experimentalists. Future generations of experiments specifically designed to detect axions – such as those looking for their conversion into photons in strong magnetic fields – could potentially confirm or refute the predictions of this model. The precision of these experiments is continuously improving, bringing us closer to potentially probing the very low-mass, weakly interacting particles predicted by axion models, thereby shedding light on both the strong CP problem and the nature of dark matter.

The research underscores the power of theoretical physics to address fundamental questions about the universe. Even when direct experimental verification is elusive, theoretical advancements can provide crucial insights and guide the direction of future research. The intricate proposal by Di Luzio, Landini, Mescia, and colleagues exemplifies how exploring complex and elegant symmetry structures can lead to profound solutions to long-standing puzzles, demonstrating the interconnectedness of different areas of fundamental physics and highlighting the potential for a unified understanding of the cosmos. This is precisely the kind of breakthrough that fuels scientific curiosity and drives the relentless pursuit of knowledge.

The implications of this research extend beyond just solving the strong CP problem and pointing towards axion dark matter. The proposed mechanism of interplay between vertical and horizontal gauge symmetries might also shed light on other outstanding puzzles in particle physics, such as the hierarchical structure of fermion masses and mixing angles, which are another set of mysteries that the Standard Model does not fully explain. A theory that can simultaneously address multiple fundamental problems is often considered more robust and indicative of underlying physical reality. This research presents an opportunity to explore these connections further and potentially develop a more complete picture of fundamental physics.

The theoretical construction is a tour de force of modern theoretical particle physics. It involves advanced group theory, the understanding of how symmetries are spontaneously broken, and the subtle interplay of quantum effects that can stabilize vacuum states. The researchers meticulously detail how the breaking of larger gauge symmetries, encompassing both vertical and horizontal aspects, cascades down to generate the specific conditions required for a high-quality Peccei-Quinn symmetry. This detailed and rigorous approach is what lends credibility to their proposal and invites detailed study by the global physics community, ensuring that the foundations of the proposed solution are robust and well-understood.

The term “high-quality” Peccei-Quinn symmetry is crucial. It refers to the fact that the symmetry breaking naturally leads to a suppression of the strong CP violation that is precisely in line with experimental observations. In some theoretical models, achieving this requires “fine-tuning” of parameters, meaning that certain constants must be set to incredibly specific values to make the theory work. This is generally considered unaesthetic by physicists. The proposed mechanism aims to avoid such fine-tuning, suggesting that the correct level of CP symmetry is a natural consequence of the underlying gauge symmetry structure, a highly desirable outcome in theoretical physics.

This work represents a significant advancement in our theoretical understanding of fundamental physics. By offering a plausible and elegant mechanism for generating a high-quality Peccei-Quinn symmetry, the authors have provided a potential solution to the strong CP problem, and in doing so, have strongly motivated the existence of axions as a dark matter candidate. This research bridges the gap between solving a conceptual puzzle in particle physics and addressing a major observational mystery in cosmology, showcasing the profound interconnectedness of these fields and the power of theoretical physics to illuminate the deepest workings of the universe. The ongoing quest for a unified understanding of reality is propelled forward by such innovative and insightful theoretical proposals.

The exploration of vertical and horizontal gauge symmetries may hint at deeper structures within the universe’s fundamental laws. These terms suggest a layered approach to symmetry in the very early universe, where different types of fundamental interactions were linked in ways that are not immediately apparent at the lower energy scales we observe today. The carefully constructed interplay of these symmetries, as proposed in the paper, is what ultimately generates the conditions necessary for the Peccei-Quinn mechanism to operate effectively, resolving the strong CP problem and pointing towards the existence of axionic dark matter.

The scientific community will undoubtedly be dissecting this research, examining its assumptions, and exploring its implications. The robustness of theoretical models that can simultaneously address multiple fundamental puzzles, like the strong CP problem and the nature of dark matter, is a strong indicator of their potential to reflect reality. This paper offers a compelling new direction for theoretical and experimental physicists alike, igniting new avenues of inquiry that could fundamentally alter our comprehension of the cosmos. The quest for knowledge is an ongoing journey, and this research represents a significant and exciting new chapter.

This research is a masterful example of how theoretical physics can tackle profound enigmas by exploring novel symmetry structures. The proposed mechanism for generating a high-quality Peccei-Quinn symmetry through the sophisticated interplay of vertical and horizontal gauge symmetries not only offers a compelling resolution to the strong CP problem but also provides a strong theoretical foundation for the existence of axions as a leading candidate for dark matter. This elegant unification of solutions to two of physics’ most persistent puzzles underscores the potential for a deeper, more interconnected understanding of the universe’s fundamental workings and serves as a powerful impetus for future experimental exploration.

Subject of Research: The resolution of the strong CP problem and the theoretical generation of axion dark matter through a novel gauge symmetry framework.

Article Title: High-quality Peccei-Quinn symmetry from the interplay of vertical and horizontal gauge symmetries.

Article References:

Di Luzio, L., Landini, G., Mescia, F. et al. High-quality Peccei-Quinn symmetry from the interplay of vertical and horizontal gauge symmetries.
Eur. Phys. J. C 86, 5 (2026). https://doi.org/10.1140/epjc/s10052-025-15175-w

Image Credits: AI Generated

DOI: https://doi.org/10.1140/epjc/s10052-025-15175-w

Keywords: Strong CP problem, Peccei-Quinn symmetry, axions, dark matter, gauge symmetry, CP violation, quantum chromodynamics, particle physics, cosmology, theoretical physics, symmetry breaking.

The team, led by R.H. Ali and X.M. Kuang, has meticulously analyzed the thermodynamic behavior of a specific type of black hole: an Anti-de Sitter (AdS) black hole imbued with a constituent of “perfect fluid dark matter.” This theoretical construct, the AdS black hole, exists in a universe with a negative cosmological constant, a concept that differs from our observed universe but is immensely useful for theoretical explorations of gravity and quantum mechanics due to its inherent properties that simplify complex calculations. The addition of perfect fluid dark matter, a hypothetical substance that behaves uniformly in all directions and is thought to constitute a significant portion of the universe’s mass-energy content, introduces a new layer of complexity and intrigue to the already mind-boggling physics of these black holes. By studying how a particle’s chaotic motion changes within this specific black hole environment, scientists can infer crucial information about the black hole’s thermodynamic state, including shifts analogous to boiling or condensation in everyday matter, but on scales so incomprehensible they challenge the imagination.

The core of this pioneering work lies in the intricate relationship between the black hole’s thermodynamic phase transitions and the Lyapunov exponent. Traditional thermodynamic systems exhibit distinct phase transitions, where a substance changes its state of matter, such as water freezing into ice or boiling into steam. These transitions are often accompanied by changes in properties like energy or entropy. This research postulates that black holes, despite their alien nature, also undergo analogous phase transitions. The novel approach is to probe these transitions not by directly measuring heat or pressure, which is impossible within a black hole, but by observing the Lyapunov exponent. A higher Lyapunov exponent signifies greater chaos and instability, while a lower one indicates a more ordered and stable state. As the black hole’s parameters, such as its mass or charge, are altered, the Lyapunov exponent will fluctuate in specific ways, mirroring the signatures of thermodynamic phase transitions with remarkable fidelity, offering an indirect yet powerful method of observation.

Furthermore, the inclusion of perfect fluid dark matter in the theoretical framework adds another dimension to the investigation, hinting at a deeper cosmic connection. Dark matter, despite its overwhelming gravitational influence, remains one of the most profound enigmas in modern physics. Its invisible nature and unknown composition have made it notoriously difficult to study. However, by observing its interaction with hypothetical black holes within the AdS spacetime, scientists might uncover clues about its fundamental properties and behavior. If the thermodynamic phase transitions of these dark matter-infused black holes are indeed directly reflected in the Lyapunov exponent, it would imply a fundamental link between gravity, thermodynamics, and the elusive nature of dark matter, potentially opening new avenues for its detection and characterization. This research daringly suggests that the secrets of dark matter might be whispered in the chaotic trajectories of particles near the edge of a black hole.

The mathematical framework employed in this study is sophisticated, involving concepts from general relativity, thermodynamics, and chaos theory. The researchers delve into the intricacies of the black hole’s metric, which describes the geometry of spacetime around it, and analyze how perturbations to this geometry, induced by the dark matter and the particle’s motion, evolve over time. The Lyapunov exponent is calculated by tracking the divergence of infinitely close initial particle trajectories, a process that, when analyzed mathematically, reveals the underlying dynamics of the system. This rigorous mathematical approach allows for precise predictions about when and how these phase transitions might occur, transforming abstract theoretical concepts into testable predictions, even if direct observational tests are currently beyond our technological capabilities for these extreme scenarios.

The implications of this research extend far beyond the purely theoretical. If the Lyapunov exponent indeed serves as a universal indicator of thermodynamic phase transitions in black holes, regardless of their specific composition, it could provide a powerful new tool for astronomers and physicists attempting to understand the evolution of the universe. Black holes are ubiquitous, from the supermassive entities at the centers of galaxies to hypothetical primordial black holes that may have formed in the early universe. Understanding their thermodynamic behavior is crucial for comprehending phenomena such as Hawking radiation, black hole mergers, and the broader cosmological evolution. This new method offers a potential pathway to probe these processes in unprecedented detail, even in the absence of direct observational data from within a black hole.

The study also touches upon the fascinating concept of phase transitions in the context of a higher-dimensional spacetime, as AdS spacetimes are often considered in dimensions greater than our familiar four spacetime dimensions. Exploring these transitions in higher dimensions can reveal emergent phenomena and symmetries that are not apparent in lower dimensions, offering new insights into quantum gravity and the fundamental nature of spacetime. The interaction of dark matter with these higher-dimensional black holes further complicates and enriches the theoretical landscape, potentially leading to unexpected discoveries about the interplay between gravity, matter, and the very fabric of reality. The mathematical elegance of these higher-dimensional models often provides profound simplifications that are otherwise intractable in our familiar four dimensions.

The perfect fluid dark matter model is a particularly compelling aspect of this research. While the exact nature of dark matter remains elusive, the perfect fluid model provides a convenient and often surprisingly accurate description of its behavior on large scales. By incorporating this model into the black hole thermodynamics, the researchers are essentially exploring the thermodynamic consequences of dark matter’s presence in extreme gravitational environments. This could lead to a deeper understanding of dark matter’s properties, such as its equation of state and its potential interactions with other fundamental forces, by observing its collective ‘phase’ changes as dictated by the black hole’s gravitational influence and its own thermodynamic fluctuations.

The concept of Lyapunov exponents, while rooted in the study of chaotic systems, has found surprising applications in diverse fields, from meteorology to economics and, now, to astrophysics. Its ability to quantify unpredictability and sensitivity to initial conditions makes it an ideal tool for probing systems that are inherently complex and difficult to model. In the realm of black holes, where direct experimentation is impossible, and theoretical modeling is fraught with challenges, the Lyapunov exponent emerges as a beacon of insight, guiding researchers through the labyrinthine complexities of these cosmic enigmas. The subtle exponential divergence of trajectories, an almost imperceptible shift in motion, carries within it the echoes of profound thermodynamic shifts.

One of the most tantalizing aspects of this research is its potential to bridge the gap between the quantum realm and the macroscopic world of black holes. Thermodynamic phase transitions are inherently macroscopic phenomena, while the motion of individual particles is governed by quantum mechanics. By using the Lyapunov exponent, which tracks the classical dynamics of particles, to infer thermodynamic properties, the researchers are effectively exploring how quantum behavior manifests in a macroscopic gravitational system. This could offer valuable insights into the long-sought unification of general relativity and quantum mechanics, a grand challenge that has occupied physicists for decades, with black holes serving as nature’s most extreme laboratories for such inquiries.

The numerical simulations and theoretical calculations involved in determining the Lyapunov exponent for these AdS black holes with perfect fluid dark matter are computationally intensive. However, the development of advanced algorithms and the increasing power of supercomputers make such investigations increasingly feasible. The satisfaction derived from unraveling these complex mathematical relationships and their physical implications is immense, pushing the boundaries of our scientific knowledge and opening up new frontiers for exploration, even if experimental verification remains a distant aspiration. Each successful calculation is a small victory in the ongoing quest to understand the universe.

The scientific community is abuzz with the possibilities presented by this research. The elegant application of chaos theory to black hole thermodynamics, coupled with the enigmatic nature of dark matter, has created a potent synergy that is likely to inspire a new wave of theoretical and potentially observational investigations. Future research may focus on exploring different types of black holes, varying the properties of the dark matter constituent, or even extending the analysis to more realistic cosmological spacetimes. The journey to decipher the universe’s deepest secrets is a continuous one, and this study represents a significant leap forward.

In conclusion, this groundbreaking work by Ali and Kuang offers a novel and powerful lens through which to view the universe’s most extreme phenomena. By harnessing the subtle dynamics of chaos, scientists are gaining unprecedented insights into the thermodynamic phase transitions of black holes and forging a surprising connection to the pervasive mystery of dark matter. This research not only deepens our understanding of fundamental physics but also serves as a testament to the ingenuity and perseverance of scientists dedicated to unraveling the universe’s grandest puzzles, proving that even in the most chaotic of environments, order and understanding can be found. The whispers of black holes, amplified by the chaos of escaping particles and permeated by the mystery of dark matter, are slowly revealing the universe’s deepest secrets.

Subject of Research: Probing thermodynamic phase transitions in Anti-de Sitter black holes with perfect fluid dark matter via the Lyapunov exponent.

Article Title: Probing thermodynamic phase transitions via Lyapunov exponent in AdS black hole with perfect fluid dark matter.

Article References:

Ali, R.H., Kuang, XM. Probing thermodynamic phase transitions via Lyapunov exponent in AdS black hole with perfect fluid dark matter.
Eur. Phys. J. C 85, 1131 (2025). https://doi.org/10.1140/epjc/s10052-025-14816-4

Image Credits: AI Generated

DOI: https://doi.org/10.1140/epjc/s10052-025-14816-4

Keywords**: Black hole thermodynamics, phase transitions, Lyapunov exponent, Anti-de Sitter black holes, perfect fluid dark matter, chaos theory, general relativity, quantum gravity.