The AdS/CFT correspondence, proposed by Juan Maldacena in the late 1990s, asserts that a gravitational theory in a volume of Anti-de Sitter spacetime is exactly equivalent, or dual, to a conformal field theory living on the boundary of that space. It is a kind of holographic dictionary: every black hole in the bulk corresponds to a hot quantum fluid on the boundary, and thermodynamic quantities such as mass, charge, temperature and entropy map onto energy, electric potential and the central charge of the field theory. The most famous example is the Hawking-Page transition, in which a black hole becomes thermodynamically favored over empty AdS space, interpreted holographically as a confinement-deconfinement transition in the boundary gauge theory. The new work, by Abhishek Baruah, Amijit Bhattacharjee and Prabwal Phukon, takes a deliberately restricted but sharp question: when a black hole is built from nonlinear electromagnetism and its entropy is deformed by generalized statistics, does the pattern of critical points on the gravity side still match the pattern on the field theory side?

The team examined three distinct classes of charged AdS black holes arising from nonlinear electrodynamics, a family of theories that modify Maxwell’s equations in strong electromagnetic fields. The first is the ModMax black hole, based on a theory remarkable for preserving both conformal invariance and electromagnetic duality while introducing a single dimensionless deformation parameter gamma. The second is a generic nonlinear electrodynamics model whose Lagrangian is suppressed by hyperbolic functions of the field invariant, producing magnetically charged black holes with a finite nonlinear field energy. The third, and richest, is the Euler-Heisenberg black hole, rooted in the genuine quantum correction to electrodynamics first calculated by Heisenberg and Euler in 1936, which captures vacuum polarization effects that become important near the horizon. In each case the nonlinear corrections alter the metric function of the black hole, and therefore reshape the temperature-entropy relation and the response functions that govern stability.

On top of these electromagnetic deformations, the authors layered a second axis of modification: the choice of entropy functional. Alongside the standard Bekenstein-Hawking area law, they employed two generalized entropies drawn from non-extensive statistical mechanics. The Renyi entropy, a one-parameter generalization of the von Neumann entropy familiar from quantum information theory, arises naturally in replica methods and probes the full entanglement spectrum. The Kaniadakis entropy, by contrast, emerges from a kappa-deformed relativistic statistical mechanics built on generalized logarithms and exponentials, providing a consistent non-Gibbsian extension of equilibrium thermodynamics. Both reduce to the Bekenstein-Hawking entropy in appropriate limits, but their deformation parameters can substantially alter the phase structure, shifting or even multiplying the critical points at which a black hole changes stability.

The analytical machinery that ties these threads together is geometrothermodynamics, a framework developed by Hernando Quevedo that applies differential geometry to the space of thermodynamic equilibrium states. Earlier metrics proposed by Weinhold and Ruppeiner suffered from a crucial flaw: they were not invariant under Legendre transformations, so different choices of thermodynamic potential yielded different curvatures. The geometrothermodynamic construction repairs this by building a Legendre-invariant metric on an extended phase space whose contact structure encodes the first law of thermodynamics. The payoff is a scalar curvature, R_GTD, with a clean physical interpretation: flat geometry corresponds to a non-interacting system such as an ideal gas, while curvature singularities mark phase transitions. For black holes, the promise is that the curvature diverges exactly where the heat capacity diverges and where the temperature-entropy curve reaches an extremum, providing a geometric, potential-independent diagnostic of criticality.

The results are strikingly consistent. For the ModMax AdS black hole with Bekenstein-Hawking entropy, the temperature profile shows two extrema at entropy values of 16.454 and 88.265, the specific heat diverges at precisely those points, and the GTD curvature develops singularities at exactly the same locations. Under Renyi entropy the two critical points persist at shifted positions, while Kaniadakis entropy introduces a third critical point that is absent in the other two descriptions. The same triple-diagnostic pattern, temperature extrema coinciding with heat capacity divergences and curvature singularities, repeats for the generic NED black hole and, most dramatically, for the Euler-Heisenberg black hole, which exhibits three critical points under Bekenstein-Hawking entropy, three under Renyi, and four under Kaniadakis statistics. The Euler-Heisenberg case is the richest of the three models, a fact the authors attribute to the higher-order QED vacuum polarization corrections that reorganize the competition between gravitational attraction and electromagnetic repulsion near the horizon.

The crucial test comes when the bulk black holes are mapped to their holographic duals. Using the restricted phase space thermodynamics framework, in which the cosmological constant is held fixed and the thermodynamic variables are reconstructed in terms of boundary quantities such as the central charge C and the CFT volume, the authors computed the energy, temperature and heat capacity of the dual field theories. The mapping is implemented through a conformal scaling factor relating bulk mass, temperature and charge to their boundary counterparts, with the central charge tied to the AdS radius and Newton’s constant. Remarkably, the number, ordering and structure of the critical points are preserved across the holographic dictionary. The ModMax black hole with two critical points maps to a CFT with two critical points; the Euler-Heisenberg system with three maps to a boundary theory with three; and the Kaniadakis-deformed descriptions carry their extra critical point from bulk to boundary in every case.

The authors are careful about what this does and does not prove. The numerical entropy values of the critical points do not coincide between bulk and boundary, and they are not expected to, since the generalized entropies are nonlinear functions of the underlying variables. What is preserved is the qualitative critical organization: the same number of thermodynamic branches, the same ordering of instabilities, and the same one-to-one correspondence between curvature singularities and heat capacity divergences. The extra Kaniadakis critical point, they argue, is statistical rather than gravitational in origin, arising because the kappa-deformed entropy functional modifies the derivatives that enter the temperature and heat capacity, rather than from any new bulk field or conserved charge. The analysis is therefore a nontrivial consistency check of bulk-boundary thermodynamics under simultaneous deformations of the gauge sector and the entropy prescription, not a full dynamical proof of AdS/CFT.

The study leaves open a tantalizing next step: extending the comparison from the locations of critical points to the near-critical scaling behavior itself, including critical exponents and universality classes. That undertaking is considerably harder within restricted phase space thermodynamics, which lacks the Van der Waals-style equation of state that makes exponent extraction straightforward in the extended phase space framework, and it would need to be performed consistently on both sides of the duality. For now, the message of this work is quietly profound. Even when the electromagnetic self-interaction is deformed by quantum vacuum polarization and the entropy law is rewritten by non-extensive statistics, the holographic mirror does not crack. The phase structure of a black hole, encoded in the curvature of an abstract thermodynamic manifold, is faithfully reflected in the thermodynamics of a quantum field theory it never touches, a consistency that continues to justify the strange and beautiful conviction that spacetime itself may be a hologram.

Subject of Research: Thermodynamic geometry and holographic consistency checks of nonlinear electrodynamics AdS black holes with generalized entropies

Article Title: Testing the AdS/CFT correspondence through thermodynamic geometry of nonlinear electrodynamics AdS black holes with generalized entropies

Article References: Baruah, A., Bhattacharjee, A., & Phukon, P. (2026). Testing the AdS/CFT correspondence through thermodynamic geometry of nonlinear electrodynamics AdS black holes with generalized entropies. The European Physical Journal C, 86(10), Article 1144. https://doi.org/10.1140/epjc/s10052-026-16330-7

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16330-7

Keywords: AdS/CFT correspondence, black hole thermodynamics, geometrothermodynamics, nonlinear electrodynamics, ModMax, Euler-Heisenberg, Renyi entropy, Kaniadakis entropy, phase transitions, holography, conformal field theory, critical points

The work, published in Nature Physics by Kaixin Ji, Yu Zhao, Ce Shen, Yidun Wan and Ling-Yan Hung, draws on some of the deepest ideas in modern condensed matter theory. The authors’ strategy does not start from spins or magnets at all. Instead, they engineer the boundary conditions of three-dimensional topological orders, exotic phases of matter whose excitations, called anyons, can carry quantum statistics that are neither bosonic nor fermionic. These topological orders are described concretely by string-net models, exactly soluble constructions introduced by Michael Levin and Xiao-Gang Wen in 2005, in which the vacuum is pictured as a tangle of fluctuating strings whose allowed patterns are dictated by algebraic data known as a fusion category.

The key innovation lies in how the critical points are created. In a topological phase, certain anyon types can undergo condensation, a process analogous to the condensation of a Bose-Einstein condensate, in which the anyon becomes part of the vacuum and other excitations are reorganized accordingly. When a single set of anyons condenses, the system typically flows from one gapped topological phase to another. The researchers instead arranged for non-commuting anyons to condense in a carefully balanced, commensurate fashion, meaning that two or more condensation channels that cannot coexist in an ordinary gapped phase are forced into competition. The tug-of-war between these incompatible orders prevents the system from settling into any gapped phase, and the resulting critical points flow in the infrared limit to conformal field theories. By tuning the relative weights of the competing condensates, the algorithm generates a lattice Hamiltonian whose low-energy behavior is precisely the desired conformal theory.

The machinery relies on a holographic device known as the strange correlator, a quantity computed as a three-dimensional path integral that maps the boundary lattice model onto the bulk topological order. In this picture, the two-dimensional critical model lives on the boundary of the three-dimensional string-net system, and the algebraic rules governing anyon fusion in the bulk translate directly into the interaction terms of the boundary model. The critical couplings, the parameter values at which the phase transitions occur, are encoded exactly in algebraic data associated with the string-net construction, specifically in the Frobenius algebras that specify which anyons condense. This means that instead of laboriously scanning parameter space numerically to hunt for critical points, physicists can read off where the transitions happen from the underlying category theory, a level of precision control that is rare in the study of strongly correlated systems.

The practical payoff is an infinite family of critical lattice models. The authors demonstrate that their procedure recovers known conformal field theories that preserve the so-called Haagerup symmetries, exotic non-invertible symmetries named after the mathematician Uffe Haagerup, whose fusion categories have intrigued both mathematicians and physicists since the 1990s. Haagerup-symmetric theories have become a testing ground for the emerging theory of categorical symmetry, in which ordinary symmetry groups are replaced by richer algebraic structures. Critical lattice models realizing these symmetries had been proposed only recently, and the new algorithm reproduces them as a special case of a much more general construction, providing independent confirmation of earlier numerical work that had reported evidence for Haagerup conformal field theories in tensor network calculations.

More strikingly, the factory does not merely recycle known results. Among the models it generates, the researchers identified three previously unknown candidate conformal field theories, critical points that had never been observed or catalogued before. These discoveries suggest that the space of two-dimensional conformal field theories is far more densely populated by accessible lattice realizations than the traditional, largely ad hoc methods of statistical mechanics had revealed. Historically, finding a lattice model for a given critical phenomenon was a matter of insight and luck, from Onsager’s solution of the Ising model to the Ashkin-Teller models studied in the early 1980s. The new algorithm replaces that serendipity with a recipe: choose a fusion category, select competing condensable algebras, and compute the resulting boundary model and its phase diagram.

The numerical verification of the construction is itself technically notable. The team developed symmetry-preserving tensor network algorithms to map out the phase diagrams of their models, coloring the parameter space by the numerically determined central charge, a fundamental invariant of a conformal field theory that measures the number of its degrees of freedom. In the phase diagrams, regions corresponding to different condensed anyon orders meet along critical lines and surfaces, and the interpolation between multiple competing condensates can be visualized in ternary diagrams representing three-condensate mixtures. The agreement between the predicted critical couplings extracted from the algebraic data and the numerical scans provides a stringent consistency check of the entire framework, and the MATLAB code and source data used to regenerate the phase diagrams have been made available with the paper.

The broader implications extend beyond two-dimensional statistical mechanics. Conformal field theories occupy a central role in high-energy theoretical physics as well, appearing as limits of quantum field theories, as building blocks of string theory, and through the AdS/CFT correspondence as dual descriptions of quantum gravity. A systematic method for discretizing conformal field theories onto lattices offers a potential route to studying them with the numerical tools of condensed matter, including tensor networks and quantum simulation. The authors and other researchers in the field have also drawn connections to topological holography and the idea that symmetries themselves can be understood as shadows of topological order, suggesting that the factory could illuminate how generalized, non-invertible symmetries emerge at quantum critical points.

The work also raises tantalizing prospects for classification. One of the great unsolved problems in theoretical physics is to classify all possible conformal field theories, a task that has proved formidable even in two dimensions where the machinery is most powerful. By establishing a structured scheme in which critical theories arise from combinatorial algebraic data, the conformal field theory factory provides a framework for discovering and potentially organizing these theories in families. If every entry in a suitable catalogue of fusion categories and condensable algebras yields a critical model, physicists may be able to enumerate, or at least systematically sample, far more of the landscape of critical behavior than ever before. For a field that has spent half a century stitching together critical phenomena one painstaking example at a time, the prospect of a factory that produces them by the dozen marks a genuine shift in method, and the three brand-new candidate theories that emerged from its first run hint at how much of that landscape still lies unexplored.

Subject of Research: An algorithm generating two-dimensional critical lattice models from competing anyon condensation in three-dimensional topological orders

Article Title: An algorithm to generate two-dimensional critical lattice models using competing anyon condensation

Article References: Ji, K., Zhao, Y., Shen, C., Wan, Y., & Hung, L.-Y. (2026). An algorithm to generate two-dimensional critical lattice models using competing anyon condensation. Nature Physics. https://doi.org/10.1038/s41567-026-03438-6

Image Credits: AI Generated

DOI: 10.1038/s41567-026-03438-6

Keywords: conformal field theory, anyon condensation, topological order, string-net models, critical phenomena, lattice models, Haagerup symmetry, phase transitions, fusion categories, tensor networks, categorical symmetry, theoretical physics