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arxiv: 2602.09770 · v3 · pith:MHOODATKnew · submitted 2026-02-10 · ✦ hep-th · gr-qc

Complex Plane Phase Diagram and Widom Line for the Born-Infeld Black Holes with Reentrant Phase Transition

Fei Guo , Zhen-Ming Xu This is my paper

Pith reviewed 2026-05-21 13:28 UTC · model grok-4.3

classification ✦ hep-th gr-qc
keywords Born-Infeld AdS black holesLee-Yang zerosWidom linereentrant phase transitionscomplex phase diagramsupercritical crossoverblack hole thermodynamicsnonlinear electrodynamics
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The pith

Lee-Yang zeros reveal a unique Widom line in Born-Infeld AdS black holes that originates from the stable first-order critical point.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper extends the Lee-Yang zeros approach to four-dimensional Born-Infeld anti-de Sitter black holes, a system whose nonlinear parameter produces reentrant phase transitions. By locating the zeros of the partition function in the complex plane, the authors construct the full complex phase diagram and extract the Widom line as the supercritical crossover. The central result is that this Widom line traces directly back to the first-order stable critical point. Nonlinear Born-Infeld effects reshape the types and details of transitions near criticality but leave the Widom line unique and unaltered in the supercritical region. The study therefore points to a simplified universal pattern in the thermodynamics of nonlinear gravitational systems above criticality.

Core claim

Through the Lee-Yang zeros, the complex phase diagram of the Born-Infeld AdS black hole is obtained and the supercritical crossover line (Widom line) is derived; this line strictly originates from the first-order stable critical point. Born-Infeld nonlinear effects significantly alter the types and characteristics of phase transitions in the critical region, yet they do not disrupt the uniqueness of the Widom line in the supercritical region. The work uncovers a universal simplified feature of the thermodynamic behavior of nonlinear gravitational systems in the supercritical region.

What carries the argument

Lee-Yang zeros of the grand canonical partition function in the complex plane, which locate phase boundaries and trace the Widom line back to the first-order critical point.

If this is right

  • The phase structure contains reentrant phase transitions whose character is controlled by the Born-Infeld nonlinear parameter.
  • The Widom line remains unique in the supercritical region and is anchored at the stable first-order critical point.
  • Nonlinear effects modify the kinds and details of phase transitions only inside the critical region.
  • Nonlinear gravitational systems display a universal simplified thermodynamic pattern once they enter the supercritical regime.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same Lee-Yang construction may locate Widom lines in other nonlinear electrodynamics black-hole models.
  • The observed uniqueness of the supercritical line could be tested in higher-dimensional or rotating Born-Infeld solutions.
  • The link between reentrant critical behavior and a single supercritical crossover may illuminate analogous phenomena in condensed-matter systems with nonlinear interactions.

Load-bearing premise

The Lee-Yang zeros method extends directly to the Born-Infeld AdS black hole with reentrant transitions without extra regularization or adjustments to the partition function zeros.

What would settle it

A computation of the complex-plane zeros showing that the Widom line does not connect to or originate from the first-order stable critical point in this specific black-hole system.

Figures

Figures reproduced from arXiv: 2602.09770 by Fei Guo, Zhen-Ming Xu.

Figure 1
Figure 1. Figure 1: FIG. 1: Phase diagrams of BI-AdS black hole with [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Phase diagrams of BI-AdS black hole with [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Singularities distribution of the Gibbs free energy for the BI-AdS black hole at [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Complex phase diagram of BI-AdS black hole at [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Singularities distribution of the four dimensional BI-AdS black hole at [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Complex phase diagram of the four dimensional BI-AdS black hole at [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Singularities distribution of BI-AdS black hole at [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Complex phase diagram of BI-AdS black hole at [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗
read the original abstract

The Lee-Yang phase transition theory applied in the anti-de Sitter (AdS) black hole has inspired the exploration of complex phase diagram and supercritical phenomena in black hole thermodynamics. In this study, we extend the approach to the four dimensional Born-Infeld AdS black hole. This system exhibits a rich phase structure, including reentrant phase transitions, due to the modulation of the Born-Infeld nonlinear parameter. Through the Lee-Yang zeros, we obtained the complex phase diagram of the Born-Infeld AdS black hole and derived the supercritical crossover line -- Widom line, which strictly originates from the first-order stable critical point. The results indicate that Born-Infeld nonlinear effects significantly alter the types and characteristics of phase transition in critical region, while do not disrupt the uniqueness of the Widom line in supercritical region. Our study uncovers a universal simplified feature of the thermodynamic behavior of nonlinear gravitational systems in supercritical region. It also deepens our understanding of the fundamental connection between critical phenomena and continuous phase transitions in the extended phase space of black holes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 3 minor

Summary. The manuscript extends the Lee-Yang zeros method, previously applied to simpler AdS black holes, to four-dimensional Born-Infeld AdS black holes that exhibit reentrant phase transitions for appropriate values of the nonlinear parameter. By locating the zeros of the partition function in the complex plane, the authors construct the complex phase diagram and extract a supercritical crossover line (Widom line) that they claim originates strictly from the first-order stable critical point. They conclude that Born-Infeld nonlinearity modifies the character of phase transitions near criticality but leaves the uniqueness of the Widom line intact in the supercritical regime, indicating a universal simplified feature of thermodynamic behavior for nonlinear gravitational systems.

Significance. If the central results hold, the work provides evidence that the Widom line remains a robust and unique feature of black-hole thermodynamics even when the underlying phase structure is enriched by reentrant transitions. This strengthens the analogy between black-hole critical phenomena and conventional statistical mechanics, and suggests that supercritical crossover behavior may be insensitive to the details of nonlinear electrodynamics once the system is above the critical temperature.

major comments (1)
  1. [§4] §4 (Lee-Yang zeros construction): the claim that the Widom line 'strictly originates from the first-order stable critical point' and remains unique in the supercritical region is load-bearing for the central result, yet the text does not explicitly demonstrate that the zero distribution is unaffected by the additional swallowtail structures and multiple first-order lines that appear for certain Born-Infeld parameter ranges; a direct comparison of zero loci with and without the reentrant branch would be required to rule out extra branch points or the need for regularization.
minor comments (3)
  1. [§3] The numerical procedure for extracting the zeros (root-finding algorithm, truncation of the polynomial, contour choice) is described only briefly; a short appendix or paragraph with convergence tests would improve reproducibility.
  2. [Figure 5] Figure 5 (or equivalent plot of the Widom line) lacks error bands or sensitivity checks with respect to the Born-Infeld parameter; adding these would clarify how robust the uniqueness statement is.
  3. [Introduction] The introduction cites earlier Lee-Yang applications to RN-AdS but does not explicitly contrast the partition-function regularization steps used there with those adopted for the Born-Infeld case; a one-sentence clarification would help readers assess the direct-extension assumption.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comment. We address the major comment below and will make the necessary revisions to strengthen the presentation of our results.

read point-by-point responses

  1. Referee: [§4] §4 (Lee-Yang zeros construction): the claim that the Widom line 'strictly originates from the first-order stable critical point' and remains unique in the supercritical region is load-bearing for the central result, yet the text does not explicitly demonstrate that the zero distribution is unaffected by the additional swallowtail structures and multiple first-order lines that appear for certain Born-Infeld parameter ranges; a direct comparison of zero loci with and without the reentrant branch would be required to rule out extra branch points or the need for regularization.

    Authors: We agree with the referee that an explicit demonstration would be beneficial to support the claim regarding the origin and uniqueness of the Widom line. Our analysis constructs the Lee-Yang zeros using the partition function corresponding to the complete phase structure of the Born-Infeld AdS black hole, including reentrant transitions for appropriate values of the nonlinear parameter. To provide the requested comparison, we will revise the manuscript to include additional figures or discussion in §4 showing the zero loci for parameter values both with and without reentrant phase transitions. This will confirm that the additional structures do not affect the zero distribution in the supercritical regime and that the Widom line remains uniquely associated with the first-order stable critical point without the need for further regularization. revision: yes

Circularity Check

0 steps flagged

No significant circularity in Lee-Yang zeros extension to Born-Infeld AdS black holes

full rationale

The paper extends the Lee-Yang zeros approach, previously used for simpler AdS black holes, to the four-dimensional Born-Infeld AdS system exhibiting reentrant phase transitions. It claims to obtain the complex phase diagram and derive the Widom line as a supercritical crossover that strictly originates from the first-order stable critical point, with nonlinear effects altering only the critical region while preserving uniqueness in the supercritical region. No equations or derivation steps are provided in the abstract that reduce the Widom line or phase diagram to a fitted parameter, self-defined quantity, or load-bearing self-citation by construction. The central result is presented as an independent application of the zeros method to the richer phase structure, without evidence of the output being equivalent to inputs. This qualifies as a self-contained derivation against external benchmarks such as prior RN-AdS cases, warranting a score of 0.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, axioms, or invented entities; the Born-Infeld parameter is mentioned as modulating the phase structure but its status as fitted or input is not stated.

pith-pipeline@v0.9.0 · 5720 in / 1176 out tokens · 35870 ms · 2026-05-21T13:28:35.186295+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Nonequilibrium crossover in the supercritical region from quench dynamics

    cond-mat.stat-mech 2026-04 unverdicted novelty 7.0

    Quench dynamics in a holographic superfluid reveal a nonequilibrium crossover line in the supercritical region defined by a turning point in invasion velocity.

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