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

The framework at the heart of the study is an extension of the familiar black hole metric in which the angular sector of the geometry is warped according to a power law, R(r) = (r/r0)^N. The exponent N acts as a continuous control parameter, analogous to pressure or temperature in a laboratory phase transition. When N is adjusted, the thermodynamic behavior of the black hole changes, and at a specific value the system undergoes a second-order phase transition. Such transitions are governed by mean-field critical exponents, the same universal fingerprints that characterize critical phenomena in magnets, fluids, and superconductors, and the amplitudes of these exponents are explicitly modulated by the deformation exponent itself.

What makes a critical point so interesting for radiation physics is the divergence of thermodynamic response functions. Near a second-order phase transition, quantities such as heat capacity become unboundedly sensitive to infinitesimal changes in state, signaling that the black hole is poised between distinct thermodynamic phases. Baghbani’s analysis shows that as the dilaton black hole approaches this criticality, the effective scattering potential that governs the propagation of quantum fields near the horizon develops a structure that can trap and amplify outgoing radiation, rather than simply filtering it away.

To quantify this effect, the study solves the Klein–Gordon equation for a scalar field propagating in the deformed background, deriving the exact effective scattering potential that controls how quantum waves tunnel through the gravitational barrier surrounding the hole. The fraction of Hawking radiation that actually escapes to infinity, known as the greybody factor, is ordinarily suppressed because the curved spacetime acts like a leaky cavity, reflecting part of the radiation back toward the horizon. Near the critical point, however, the potential develops a pronounced resonant feature, and Baghbani proposes a phenomenological Breit–Wigner resonance model to capture the resulting enhancement of the greybody factor. This is the same mathematical form used to describe resonant scattering in nuclear and particle physics, suggesting a deep analogy between black hole radiance and the resonance phenomena familiar from laboratory experiments.

The dynamical side of the story is told by quasinormal modes, the characteristic ringing frequencies at which a perturbed black hole settles back to equilibrium. Working in the eikonal limit, where the perturbations have short wavelengths, the study establishes a correspondence between these modes and the unstable photon orbit, the precarious circular light trajectory that hovers just outside the horizon. Third-order WKB computations of the quasinormal mode spectra reveal that the deformation exponent N modulates both the oscillation frequency of the ringdown and the Lyapunov damping rate that controls how quickly the ringing decays. In other words, the same geometric parameter that drives the thermodynamic phase transition also reshapes the black hole’s gravitational-wave signature, offering a potential observational handle on the underlying physics.

A crucial consistency check comes from the limits of the model. When the deformation exponent N approaches zero from below and the dilaton parameter α goes to zero, all thermodynamic, dynamical, and radiative quantities reduce smoothly to the Reissner–Nordström–AdS limit, the well-understood solution describing a charged black hole in anti-de Sitter space. This means the exotic behavior near criticality is not an artifact of the deformation but a genuine feature that interpolates between known black hole physics and a new regime of critical behavior. The recovery of established results in appropriate limits is an important sanity test for any proposal in gravitational theory.

The broader significance of the work lies in its suggestion that quantum gravitational effects, normally hopelessly beyond experimental reach, might be amplified by manipulating the geometry of spacetime itself. Hawking radiation is far too weak to detect for astrophysical black holes, but in analog systems and in highly controlled theoretical backgrounds, the interplay between geometry and quantum fields becomes tractable. If geometric deformation provides a tunable knob for the greybody factor and the radiation spectrum, it opens a pathway for probing how quantum mechanics and general relativity conspire at horizons, a question at the very frontier of theoretical physics.

The study also connects to an active body of research on black hole phase transitions and thermodynamic geometry. Critical phenomena in black hole thermodynamics have been explored extensively in charged and rotating solutions, in extended thermodynamics where the cosmological constant is treated as pressure, and in Ruppeiner-style geometric formulations of statistical mechanics. What distinguishes the present analysis is the treatment of a pure geometric deformation exponent as an active control parameter, rather than varying charge, rotation, or background curvature. This reframing suggests that the landscape of black hole phases is richer than previously appreciated and that some of its most dramatic features occur where response functions diverge.

Caveats remain. The resonant enhancement of the greybody factor is currently a phenomenological model rather than a direct numerical computation, and the author is explicit that direct numerical evaluation of the greybody factor is required to confirm the predicted enhancement. Moreover, the four-dimensional dilaton black hole with a deformed angular sector is a theoretical construction, and whether configurations of this type exist in nature or can be realized in analogue systems is an open question. Quasinormal mode calculations at third WKB order likewise carry controlled but finite uncertainties that full numerical evolution would help pin down.

Even so, the picture that emerges is striking. A single geometric parameter governs a second-order phase transition, reshapes the scattering potential for quantum fields, breathes resonant structure into the escaping radiation, and rewrites the ringdown spectrum, all while recovering known black hole physics in the appropriate limits. As gravitational-wave detectors grow more sensitive and analogue gravity experiments grow more sophisticated, the idea that black hole radiation can be critically enhanced by warping geometry may evolve from an elegant calculation into a guiding principle for the hunt for quantum gravity. The study, published as Volume 58, article 102 of General Relativity and Gravitation, was received in April 2026, accepted in late August 2026, and published on 1 September 2026.

Subject of Research: Critical enhancement of Hawking radiation in geometrically deformed dilaton black holes

Article Title: Critical enhancement of Hawking radiation in geometrically deformed dilaton black holes

Article References: Baghbani, R. (2026). Critical enhancement of Hawking radiation in geometrically deformed dilaton black holes. General Relativity and Gravitation, 58(9), Article 102. https://doi.org/10.1007/s10714-026-03607-1

Image Credits: AI Generated

DOI: 10.1007/s10714-026-03607-1

Keywords: Hawking radiation, dilaton black holes, geometric deformation, black hole thermodynamics, phase transition, critical phenomena, greybody factor, quasinormal modes, Breit-Wigner resonance, photon orbit, eikonal limit, quantum gravity