The system at the center of the analysis is the Euler-Heisenberg anti-de Sitter (AdS) black hole, a solution of general relativity coupled to a nonlinear version of electromagnetism. Nonlinear electrodynamics arises naturally when quantum effects modify Maxwell’s classical theory, and the Euler-Heisenberg Lagrangian is the canonical example, originally derived to describe how vacuum fluctuations soften extremely strong electric fields. In the black hole spacetime, the strength of these nonlinear corrections is controlled by a single parameter, usually denoted a, which appears in the metric function as a term proportional to a times the fourth power of the charge divided by the sixth power of the radius. That seemingly modest correction turns out to reshape the entire thermodynamic landscape, altering the equation of state and generating qualitatively different phase behavior depending on the parameter’s value.

The framework in which this behavior is interpreted is known as extended phase space thermodynamics, or black hole chemistry. In this picture, the cosmological constant of the spacetime is no longer treated as a fixed background quantity but as a genuine thermodynamic variable identified with pressure. Once pressure enters the first law, charged AdS black holes begin to mimic van der Waals fluids, exhibiting first-order phase transitions between small and large black hole phases that terminate at critical points, exactly as liquid and gas phases terminate at the critical point of a real fluid. For the Euler-Heisenberg black hole, the parameter a modulates this analogy: for negative values the system shows the standard van der Waals-like transition, for intermediate values up to a threshold set by the charge squared it develops reentrant phase transitions, and for sufficiently large values the transition disappears altogether, leaving a single thermodynamic phase.

The new study focuses on a regime where the mathematics becomes especially intricate. Solving the standard criticality conditions, that the first and second derivatives of the Hawking temperature with respect to the horizon radius vanish simultaneously, yields not one but two distinct sets of critical parameters. At fixed pressure, the first critical point sits at a horizon radius of about 0.59 with a critical temperature near 0.079, while the second lies at a radius of roughly 4.46 with a temperature near 0.024. Below these critical values, the thermodynamic behavior splits into four separate black hole branches, which the authors label the Smallest, Small, Intermediate, and Large black hole phases. The temperature curve possesses three extrema that carve the parameter space into these four sectors, and the specific heat diverges at three locations, marking the boundaries between branches where stability flips sign. Branches with positive specific heat are locally stable; those with negative specific heat are unstable and cannot persist against small fluctuations.

Even more striking is what happens when the pressure is tuned. As pressure increases, the two critical points migrate toward one another and eventually coalesce at a unique set of parameters, a horizon radius of about 0.296, a charge of roughly 0.148, and a pressure near 0.171. At this degenerate critical point, the first three derivatives of the Hawking temperature vanish simultaneously, producing a higher-order stationary inflection point in the temperature profile. The four-phase structure collapses into just two branches, one stable and one unstable, and the separate free-energy curves that distinguished the multiple phases merge into a simplified structure. The degenerate point therefore acts as the merger of two independent criticalities, a higher-order critical phenomenon that has no analogue in the ordinary van der Waals system.

To probe what happens above and beyond these critical points, the authors turned to the Lee-Yang theory of phase transitions, formulated in 1952 by Tsung-Dao Lee and Chen-Ning Yang. The central insight of that theory is that phase transitions are encoded in the zeros of a system’s partition function, the mathematical object that sums over all possible configurations. For finite systems these zeros live in the complex plane, but as the system grows toward the thermodynamic limit they can pinch the real axis, and where they do, the Gibbs free energy becomes non-analytic: a phase transition. In black hole thermodynamics, the partition function is related to the Euclidean gravitational action, and the Gibbs free energy equals minus the temperature times the logarithm of the partition function. Consequently, the Lee-Yang zeros of the black hole system can be located by finding where the specific heat diverges, since the specific heat is proportional to the second derivative of the free energy.

Crucially, the complex horizon radius in this construction is not a physical claim that horizons occupy imaginary positions in spacetime. It is an analytic continuation, a mathematical device for tracking where the thermodynamic functions develop singularities. In the critical region, the relevant singularities sit on the real axis, corresponding to spinodal boundaries where the specific heat blows up. Once the system enters the supercritical regime, the singularities lift off the real axis and travel as complex-conjugate pairs. The authors found that in the two-critical-point regime, the singularities organize into two separate families, each anchored at one of the critical points, and that as the degenerate point is approached, these families coalesce into a single configuration whose branches intersect the real axis at exactly the degenerate critical radius.

Projecting these complex singularity trajectories onto the real temperature-pressure plane produces the Widom line, a crossover boundary that divides the supercritical region into sectors with liquid-like and gas-like character in ordinary fluids, or small-black-hole-like and large-black-hole-like character in gravitational ones. The surprise lies in what the projections mean for the Euler-Heisenberg system. At the degenerate critical point, no conventional coexistence curve exists, because there are not two stable phases with equal free energy to coexist. Yet a well-defined Widom line still emerges from the complex singularity structure, and it does real thermodynamic work: the specific heat changes sign across it, so the projected line acts as an effective stability boundary separating locally stable from locally unstable sectors of the supercritical parameter space. In the two-critical-point regime, the situation becomes richer still. One critical point carries the conventional story, with a coexistence curve terminating on it and a Widom line continuing that curve into the supercritical region. The second critical point, by contrast, has no coexistence line at all, but its complex singularities still generate a second, independent Widom line. Two qualitatively different crossover structures thus coexist within the same black hole system, demonstrating that supercritical crossovers need not share a single geometric origin.

The authors also verified the framework’s consistency in limiting cases. When the system possesses a single ordinary critical point, the Lee-Yang singularities intersect the real axis exactly once and reproduce the textbook picture of one coexistence curve and one Widom line, here separating intermediate and large black hole regimes, with the coexistence curve bending in accordance with the Clapeyron relation as entropy and thermodynamic volume vary along it. And when the Euler-Heisenberg parameter is large enough that no criticality exists, the complex-conjugate singularity trajectories remain forever detached from the real axis, the Gibbs free energy stays analytic, and no phase transition signature appears. The approach of Lee-Yang zeros toward the real axis is thus a faithful diagnostic: when they reach it, criticality follows; when they do not, the thermodynamic landscape remains smooth.

The broader significance of the work is twofold. First, it establishes the complex phase diagram as a practical tool for mapping black hole thermodynamics beyond the critical point, a regime that has received far less attention than the subcritical territory where most known transitions live. Second, and perhaps more conceptually provocative, it shows that Widom lines can exist and carry physical meaning even in the complete absence of a first-order coexistence curve, suggesting that the complex singularity structure of the free energy encodes crossover phenomena that conventional real-valued analysis misses entirely. As theoretical physicists continue to catalog the phase structures of increasingly exotic black hole solutions, from Gauss-Bonnet to Born-Infeld to noncommutative geometries, the message of this study is that the richest information may reside not on the real thermodynamic plane but in the complex manifold just beyond it, where the shadows of vanished phase transitions still leave visible traces.

Subject of Research: Supercritical thermodynamics and Lee-Yang phase transition theory of Euler-Heisenberg AdS black holes

Article Title: Complex phase structure and Widom line for Euler–Heisenberg black holes

Article References: Awal, M. B., & Phukon, P. (2026). Complex phase structure and Widom line for Euler–Heisenberg black holes. The European Physical Journal C, 86(10), Article 1146. https://doi.org/10.1140/epjc/s10052-026-16405-5

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16405-5

Keywords: black hole thermodynamics, Euler-Heisenberg black holes, AdS black holes, Lee-Yang zeros, Widom line, phase transitions, critical points, supercritical regime, nonlinear electrodynamics, extended phase space, Gibbs free energy, spinodal lines