Physicists have made substantial progress in classifying quantum phases for isolated systems, but open, non-equilibrium systems—those that freely interact and exchange information with their environment—have presented stubborn technical challenges. Without a reliable classification scheme, our understanding of fundamentally new quantum behaviors has been limited, along with our ability to harness them for quantum-computing applications. In a paper published in the journal Physical Review X on October 2, 2026, physicists at the Anthony J. Leggett Institute for Condensed Matter Theory introduced a framework for classifying non-equilibrium quantum phases of matter, generalizing principles that govern closed systems and overcoming the shortcomings of existing classification methods.

To appreciate the significance of the advance, it helps to recall how phases have traditionally been sorted. Ordinary phases are generally classified by their symmetries. A liquid looks roughly the same from any direction and therefore possesses a high degree of symmetry, whereas a crystalline solid exhibits symmetry only along well-defined axes—a difference that immediately signals the two are distinct phases. This paradigm, pioneered by Lev Landau in the 1930s, has extraordinary explanatory power, describing everything from solids, liquids, and gases to magnets and even superconductors.

Since the 1980s, however, physicists have realized that many phases cannot be explained by symmetry alone. They also require topology, a branch of mathematics that studies fundamental, global properties of shapes while ignoring local, small-scale details. In the oft-cited example, a coffee cup can be mathematically massaged, or continuously deformed, into a donut, showing that the two shapes are globally identical and belong to the same class. Each class is characterized by topological invariants—special numbers, such as the number of holes a shape possesses, that do not change under continuous deformation. Producing a genuinely new topological shape requires a discontinuous change such as tearing, an operation that is strictly banned in topology. Topological phases of matter are distinguished by their invariants in exactly the same way, maintaining their order even when locally perturbed by external interactions. Physically, they arise when temperatures are lowered so far that quantum fluctuations, once masked by thermal jostling, emerge to produce system-wide quantum entanglement.

In such phases, information is stored nonlocally, smeared out across the entire system rather than pinned to any particular location. Particle-like excitations called anyons, which do not fall into the familiar boson-fermion paradigm, can also arise—a phenomenon physicists are only beginning to understand. Because of their resilience against local disturbances, topological phases are considered promising building blocks for quantum-computing technologies, which are notoriously vulnerable to environmental noise. But classifying these phases has depended on a convenient idealization. Phases in closed quantum systems, isolated from their environment, are typically described using tractable quantum states called pure states, and classification conventionally rests on the structure of their gapped Hamiltonians—mathematical objects that encode a state’s energies and impose energy gaps between states.

Illinois Physics Professor Jong Yeon Lee emphasized the core principle behind any good definition of a phase. The most important idea, he explained, is that a phase should be stable: it should be defined so that if you perturb the state slightly, it still stays in the same phase. For pure states, this notion is well defined—two states belong to the same phase if their local parent Hamiltonians can be connected without closing the energy gap, while an unavoidable gap closing signals a phase transition. The trouble is that pure states are often idealizations. In the real world, quantum systems interact with their surroundings and degrade, or decohere, turning into unpredictable statistical mixtures called mixed states. Unlike pure states, out-of-equilibrium mixed states do not have Hamiltonians at all, so the conventional approach to classification simply does not work for them.

Luckily, an alternative scheme exists: entanglement bootstrapping. In this procedure, one concocts stability criteria that define fixed points—quantities of a system that look the same at different length scales. Physicists hunt for a system’s fixed points by zooming out to large length scales, a process known as coarse-graining, to observe how the system’s parameters change. As Lee elaborated, each point in the phase diagram converges to a fixed point as one coarse-grains further and further, and phases of matter can be thought of as perturbations away from these fixed points. Such points serve as anchors for defining phases: once identified, one can move away from them and look for regions of stable quantum states, which can be collectively defined as phases. This bootstrapping strategy has found considerable success in classifying pure-state phases, and Lee’s team wondered whether the same approach could be extended to mixed states—an effort that gained momentum when Illinois Physics Postdoctoral Fellow Bowen Shi joined the group.

Lee described the entanglement-bootstrap program as a complementary way of looking at the same physics. Instead of starting from a Hamiltonian, it asks how much topological order can be reconstructed directly from the entanglement structure of a given quantum state. This philosophy is especially useful for the mixed-state problem, suggesting that a small set of information-theoretic properties can play the same role that the Hamiltonian plays for pure states. To find fixed points for mixed-state phases, the researchers devised three conditions. The first, M0, ensures stability by requiring that any two physically separated regions of matter, A and C, do not affect each other much; otherwise, perturbations could exploit long-range correlations to propagate throughout the system and destabilize the phase, much like a highway pileup that spreads from car to car when traffic is too jam-packed. A second condition, P0, demands that if information encoded in a local region C is lost or corrupted, its surrounding neighborhood B can recover it—a requirement reflecting the fact that topological information is stored nonlocally, spread across the system through long-range entanglement. Finally, condition M1 imposes mathematical technicalities on quantum states to ensure they are tractable enough to capture the right topological physics.

These three conditions hold at all length scales and together define fixed points, so any quantum state satisfying all three simultaneously is stable. Notably, if the scheme is restricted to pure states, the three conditions become equivalent to those used in pure-state bootstrapping, meaning the new framework includes pure-state classification as well. The researchers also derived several important quantities, collectively known as topological data, to distinguish and characterize different fixed points. These data not only measure the topological information content of the fixed points but also act as topological invariants, providing a way to label phases once they are defined. To complete the bootstrapping procedure and define whole phases, the researchers allowed their stability criteria to relax: states near fixed points need not satisfy the conditions exactly, only approximately. Their definition states that if one can build an intermediate boundary between two mixed states at which the deviation from the stability criteria drops off exponentially fast under coarse-graining, then the states belong to the same phase. In other words, if two states look the same when you zoom out—except for an exponentially small difference—they are essentially the same phase. Numerical implementation of coarse-graining confirmed that the deviation indeed decays exponentially, showing that the states are stable enough to define phases. Significantly, the definition is compatible with the topological data: because of topological invariance, two mixed states with different values of a topological quantity must belong to different phases, offering a quick and reliable test for labeling distinct phases without re-establishing the stability criteria from scratch.

How does this scheme compare with existing methods? The leading approach for classifying mixed-state phases relies on the idea of a finite-depth local channel, or FDLC, a kind of communication link between two quantum states. According to that approach, two states belong to the same phase if an FDLC maps one to the other and another FDLC maps it back. Upon closer inspection, however, this approach breaks down. Lee’s team gives an example of two states—a generic product state and the maximally dephased toric code—that physically belong to different phases even though the FDLC definition incorrectly classifies them as the same phase. As Lee noted, the FDLC approach does not necessarily preserve topological structure; researchers appear to have overlooked this subtlety when naively extending the idea to mixed states. Mixed-state bootstrapping, by contrast, correctly categorizes the product state and the toric code into distinct phases, in exact agreement with what is observed physically. Lee explained that the method diagnoses what goes wrong: the channel changes the underlying topological structure, violating one of the stability criteria, so bootstrapping delivers intrinsic diagnostics rather than defining a phase solely through the existence of a particular preparation protocol.

With this new classification scheme, the researchers have surmounted major hurdles in frontier condensed matter theory, most notably the long-standing inability to clearly distinguish quantum phases that are genuinely different. And they are not finished. Lee’s team is actively hunting for more topological data to complete the framework, since computing certain invariants exactly is not always easy or even possible, and states sharing the same invariant value could still belong to different phases. The team also aims to incorporate mixed-state topological order into actual experimental devices and to explore specific phases and phase transitions, particularly near critical points where different types of correlations blow up. What excites Lee most, he said, is that the work points toward a new way of defining phases of matter directly from how quantum information is organized in a state, both in and out of equilibrium. He acknowledged that the bootstrap framework is not yet universal—it does not naturally capture fracton phases, for example—and he views that limitation as an important clue, hoping ultimately to develop a broader framework encompassing these more exotic forms of quantum matter as well. The research was primarily supported by faculty startup funds from the University of Illinois Urbana-Champaign, with additional support from the Taiwan-UIUC Scholarship Program and the Elite Dream Project Grant of the Veterans and Dependents Foundation, the Perimeter Institute for Theoretical Physics, the National Science Foundation under Award No. PHY-2337931, and the IBM-Illinois Discovery Accelerator Institute, along with computing resources from the Illinois Campus Cluster Program in conjunction with the National Center for Supercomputing Applications.

Subject of Research: Classification of non-equilibrium mixed-state topological phases of matter

Article Title: Classifying non-equilibrium phases of matter

Article References: Classifying non-equilibrium phases of matter. (n.d.). Original publication

Image Credits: AI Generated

DOI: Not provided

Keywords: quantum phases of matter, non-equilibrium systems, mixed states, topological order, entanglement bootstrapping, condensed matter theory, anyons, quantum entanglement, quantum computing, coarse-graining, topological invariants, decoherence

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